Diophantine Approximation and Dirichlet Series [2 ed.] 9789811593505, 9789811593512, 9789386279828

Table of contents :
Preface
Acknowledgments
Contents
About the Authors
1 A Review of Commutative Harmonic Analysis
1.1 The Haar Measure
1.1.1 Locally Compact Abelian Groups
1.1.2 Existence and Properties of the Haar Measure
1.2 The Dual Group and the Fourier Transform
1.2.1 Characters and the Algebra L1(G)
1.2.2 Topology on the Dual Group
1.2.3 Examples and Basic Facts
1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems
1.3.1 An Abstract Theorem
1.3.2 Applications to Harmonic Analysis
1.4 The Inversion and Plancherel Theorems
1.4.1 The Inversion Theorem
1.4.2 The Plancherel Theorem
1.4.3 A Description of the Wiener Algebra
1.4.4 A Basic Hilbert-Type Inequality
1.5 Pontryagin's Duality Theorem and Applications
1.5.1 The Pontryagin Theorem
1.5.2 Topological Applications of Pontryagin's Theorem
1.6 The Uncertainty Principle
1.6.1 The Uncertainty Principle on the Real Line
1.6.2 The Uncertainty Principle on Finite Groups
1.7 Exercises
References
2 Ergodic Theory and Kronecker's Theorems
2.1 Elements of Ergodic Theory
2.1.1 Basic Notions in Ergodic Theory
2.1.2 Ergodic Theorems
2.2 The Kronecker Theorem
2.2.1 Definitions
2.2.2 Statement and Proof of the Main Theorem
2.2.3 A Useful Formulation of Kronecker's Theorem
2.3 Distribution Problems
2.3.1 Distribution in mathbbTd
2.3.2 Powers of an Algebraic Number
2.4 Towards Infinite Dimension
2.5 Exercises
References
3 Diophantine Approximation
3.1 One-Dimensional Diophantine Approximation
3.1.1 Historical Survey
3.1.2 How to Find the Best Approximations?
3.1.3 Classification of Numbers
3.1.4 First Arithmetical Results
3.2 The Gauss Ergodic System
3.3 Back to Transcendence
3.3.1 Metric Results
3.3.2 Simultaneous Diophantine Approximations
3.4 Exercises
References
4 General Properties of Dirichlet Series
4.1 Introduction
4.2 Convergence Abscissas
4.2.1 The Bohr–Cahen Formulas
4.2.2 The Perron–Landau Formula
4.2.3 The Holomorphy Abscissa
4.2.4 A Class of Examples
4.2.5 A More Intricate Dirichlet Series
4.2.6 Automatic Dirichlet Series
4.3 Products of Dirichlet Series
4.4 Bohr's Abscissa via Kronecker's Theorem
4.4.1 Bohr's Point of View
4.4.2 The Bohr Inequalities
4.4.3 A Wiener Lemma for Dirichlet Series
4.5 A Theorem of Bohr and Jessen on Zeta
4.6 Exercises
References
5 Probabilistic Methods for Dirichlet Series
5.1 Introduction
5.2 A Multidimensional Bernstein Inequality
5.3 Random Polynomials
5.3.1 Maximal Functions in Probability
5.3.2 The Sub-Gaussian Aspect of Rademacher-Type Variables
5.3.3 The Kahane Bound for Random Trigonometric Polynomials
5.3.4 Random Dirichlet Polynomials
5.4 The Proof of Bohnenblust–Hille's Theorem
5.4.1 An Elementary Version
5.4.2 A Sharp Version of the Bohnenblust–Hille Theorem
5.5 Exercises
References
6 Hardy Spaces of Dirichlet Series
6.1 Definition and First Properties
6.1.1 The Origin of the Spaces mathcalHinfty and mathcalH2
6.1.2 A Basic Property of mathcalHinfty
6.2 The Banach Space mathcalHinfty
6.2.1 The Banach Algebra Structure of mathcalHinfty
6.2.2 Behaviour of Partial Sums
6.2.3 Some Applications of the Control of Partial Sums
6.3 Additional Properties of mathcalHinfty
6.3.1 An Improved Montel Principle
6.3.2 Interpolating Sequences of mathcalHinfty
6.4 The Hilbert Space mathcalH2
6.4.1 Definition and Utility
6.4.2 The Embedding Theorem
6.4.3 Multipliers of mathcalH2
6.5 The Banach Spaces mathcalHp
6.5.1 A Basic Identity
6.5.2 Definition of Hardy-Dirichlet Spaces
6.5.3 A Detour Through Harmonic Analysis
6.5.4 Helson Forms
6.5.5 Harper's Breakthrough and Two Consequences
6.6 A Sharp Sidon Constant
6.6.1 Reformulation of Bohr's Question in Terms of mathcalHinfty
6.6.2 Symmetric Multilinear Forms
6.6.3 The Claimed Sharp Upper Bound
6.6.4 A Refined Inclusion
6.7 Exercises
References
7 Voronin-Type Theorems
7.1 Introduction
7.1.1 A Reminder About Zeta and L-Functions
7.1.2 Universality
7.2 Hilbertian Results
7.2.1 A Hilbertian Density Criterion
7.2.2 A Density Result in Bergman Spaces
7.3 Joint Universality of the Sequence (λN)
7.4 A Generalized and Uniform Carlson Formula
7.4.1 Estimates on the Gamma Function
7.4.2 The Carlson Formula
7.5 Joint Universality of the Singleton λ= (L(s, χj))
7.5.1 Notations and the Idea of Proof of Theorem 1.2
7.5.2 Details of Proof
7.6 Exercises
References
8 Composition Operators on the Space mathcalH2 of Dirichlet Series
8.1 Introduction
8.2 The Main Theorem
8.3 The Arithmetic Theorem
8.4 Twisting
8.4.1 Definition and Uniform Convergence
8.4.2 Mapping Properties
8.4.3 Twisting and Composition
8.4.4 Twisting and Probability
8.4.5 Twisting and Topology
8.5 Integral Representation and Embedding
8.5.1 Integral Representation
8.5.2 Embedding Results for mathcalH2
8.6 Proof of the Main Theorem
8.6.1 Proof of Necessity in Theorem 8.6.1
8.6.2 Proof of Sufficiency in Theorem 8.6.1
8.7 Compact Operators and Approximation Numbers
8.7.1 Definitions
8.7.2 First Properties of Approximation Numbers
8.7.3 The Multiplicative Inequalities of H. Weyl
8.8 A Lower Bound
8.9 Upper Bounds
8.10 The Case of mathcalHp-spaces
8.10.1 Reminder
8.10.2 Failure of Embedding for 0

Citation preview

Texts and Readings in Mathematics 80

Hervé Queffelec Martine Queffelec

Diophantine Approximation and Dirichlet Series Second Edition

Texts and Readings in Mathematics Volume 80

Advisory Editor C. S. Seshadri, Chennai Mathematical Institute, Chennai, India Managing Editor Rajendra Bhatia, Ashoka University, Sonepat, India Editorial Board Manindra Agrawal, Indian Institute of Technology, Kanpur, India V. Balaji, Chennai Mathematical Institute, Chennai, India R. B. Bapat, Indian Statistical Institute, New Delhi, India V. S. Borkar, Indian Institute of Technology, Mumbai, India Apoorva Khare, Indian Institute of Sciences, Bangalore, India T. R. Ramadas, Chennai Mathematical Institute, Chennai, India V. Srinivas, Tata Institute of Fundamental Research, Mumbai, India Technical Editor P. Vanchinathan, Vellore Institute of Technology, Chennai, India

The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars and teachers would find this book series useful. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Books in this series are co-published with Hindustan Book Agency, New Delhi, India.

More information about this series at http://www.springer.com/series/15141

Hervé Queffelec Martine Queffelec •

Diophantine Approximation and Dirichlet Series Second Edition

123

Hervé Queffelec Department of Mathematics University of Lille Lille, France

Martine Queffelec Department of Mathematics University of Lille Lille, France

ISSN 2366-8717 ISSN 2366-8725 (electronic) Texts and Readings in Mathematics ISBN 978-981-15-9350-5 ISBN 978-981-15-9351-2 (eBook) https://doi.org/10.1007/978-981-15-9351-2 Jointly published with Hindustan Book Agency The print edition is not for sale in India. Customers from India please order the print book from: Hindustan Book Agency. ISBN of the Co-Publisher’s edition: 978-93-86279-82-8 1st edition: © Hindustan Book Agency (India) 2013 2nd edition: © Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The birth act of the analytic theory of the Dirichlet series AðsÞ ¼

1 X

an ns

ð1Þ

n¼1

can be rightly claimed to be the Dirichlet Arithmetic Progression Theorem. In that case, the arithmetical function is n 7! an , the indicator function of the integers n congruent to q mod b for some given pair ðq; bÞ of coprime integers, and its properties are reflected in a subtle way in the “analytic” properties of the function A, although for the Dirichlet the variable s remains real. Later on, in the case of the zeta function, Riemann in his celebrated Memoir allowed complex values for s and opened the way to the proof by Hadamard and de la Vallée-Poussin of the Prime Number Theorem. The utility of those Dirichlet series for the study of arithmetical functions and of their summatory function X A ðxÞ ¼ an nx

was widely confirmed during the first half of the twentieth century, with the expansion of Tauberian theorems, including those related to Fourier and harmonic analysis, in the style of Wiener, Ikehara, Delange, etc. The hope was of course that progress on those series would imply progress on the distribution of primes, and perhaps a solution to the Riemann hypothesis, the last big question left open in Riemann’s Memoir.

v

vi

Preface

A parallel aspect also appeared in the work of H.Bohr, where the series (1) and their generalization 1 X

an e‚n s

ð2Þ

n¼1

began to be studied for themselves. In particular, Bohr proved a fundamental theorem relating the uniform convergence of a Dirichlet series (and therefore almost-periodicity properties) and the boundedness of its sum A in some half-plane. This naturally led him to his famous question on the maximal gap between abscissas of uniform and absolute convergence. Surprisingly, this question turned out to be very deep and led him to develop fairly sophisticated tools of other branches, either of complex or harmonic analysis or of diophantine approximation, through the Kronecker approximation theorem (what is called nowadays the Bohr point of view: the unique factorization in primes is seen as the linear independence of the logarithms of those primes). The central importance of this theorem in the theory of the Dirichlet series was quickly recognized by him. A solution to his question, found by Bohnenblust and Hille in a famous paper of the Annals, was obtained along the lines suggested by Bohr. Many notions of harmonic analysis (Littlewood’s multilinear inequality, p-Sidon sets, Rudin–Shapiro polynomials, etc.) were underlying in that work. The Kronecker theorem (simultaneous, non-homogeneous, approximation) points at two other aspects: on the one hand, at ergodic theory through its formulation and proof, which will be used again in the final chapter on universality, and on the other hand, at diophantine approximation, which as a consequence is very present in the book. In particular, a thorough treatment of the continued fraction expansion of a real number is presented, as well as its ergodic aspects through the Gauss map (ergodic theory again). This in turn allows a sharp study of the abscissas of convergence of classes of the Dirichlet series, which extends a previous study by Hardy–Littlewood for the (easier) case of the Taylor series. The simultaneous approximation is still not well understood, except in some cases as the sequence of powers of some given real number, like the Euler basis e, through the use of Padé approximants. A detailed presentation of those approximants and their applications to a streamlined proof of the transcendency of e is given in Chap. 3. Needless to say, the hope of solving Riemann’s hypothesis through the study of series (1) has not been completely met, in spite of many efforts. But along the lines of Bohr, Landau (also S.Mandelbrojt as concerns the series (2)) and others, those series continued to be studied for their own sake. Then came a period of relative lack of interest for that point of view, from 1960 to 1995, with several noticeable exceptions, among which was the Voronin theorem (1975) which emphasized the universal role of the zeta function, even if it made no specific progress on the Riemann hypothesis. It seems that the subject was rather suddenly revived by an important paper of Hedenmalm, Lindqvist, and Seip (1997), where several of the forgotten properties of the Dirichlet series were successfully revisited for the

Preface

vii

solution of a Hilbertian problem dating back to Beurling (Riesz character of a system of dilates of a given function), and new Hilbert and Banach spaces of the Dirichlet series were defined and studied. That paper stimulated a series of other, related, works, and this is part of those works, dating back to the past 30 years, which is exposed in those pages. The aim of this introductory book, which has the ambition of being essentially self-contained, is therefore twofold: (1) On the one hand, the basic tools of diophantine approximation, ergodic theory, harmonic analysis, probability, necessary to understand the fundamentals of the analytic theory of the Dirichlet series, are displayed in detail in the first chapters, as well as general facts about those series, and their products. (2) On the other hand, in the last two chapters, especially in Chap. 6 more recent and striking aspects of the analytic theory of the Dirichlet are presented, as an application of the techniques coined before. One fascinating aspect of that theory is that it touches many other aspects of number theory (obviously!) but also of functional, harmonic or complex analysis, so that its detailed comprehension requires a certain familiarity with several other subjects. Accordingly, this book has been divided in seven chapters, which we now present one by one. 1. Chapter 1 is a review of harmonic analysis on locally compact abelian groups, with its most salient features, including the Haar measure, dual group, Plancherel and Pontryagin’s theorems. It also insists on some more recent aspects, like the uncertainty principle for the line or a finite group (Tao’s version) and on the connection with the Dirichlet series (embedding theorem of Montgomery and Vaughan). 2. Chapter 2 presents the basics of ergodic theory (von Neumann, Oxtoby and Birkhoff theorems) with special emphasis on the applications to the Kronecker theorem (whose precised forms will be of essential use in Chap. 7), to one or multi-dimensional equidistribution problems and also to some classes of algebraic numbers (Pisot and Salem numbers). 3. Chapter 3 deals more specifically with diophantine approximation (continued fractions) in relationship with ergodic theory (Gauss transformation, which is proved to be strong mixing) and aims at giving a classification of real numbers according to their rate of approximation by rationals with controlled denominator. This classification is given by a theorem of Khintchine, fully proved here. As a corollary, the transcendency of the Euler basis e is completely proved. 4. Chapter 4 presents the basics of the general Dirichlet series of the form (1), with the Perron formulas and the way to compute the three abscissas of simple, uniform, absolute convergence, and with some comments and examples on a fourth abscissa (the holomorphy abscissa). Several classes of examples are examined in detail, including the series

viii

Preface 1 X ns jjnhjj n¼1

5.

6.

7.

8.

ð3Þ

according to the diophantine properties of the real number h. An exact formula for the abscissa of convergence of this series is given in terms of the continued fraction expansion of h. A section on so-called “automatic Dirichlet series”, in connection with “automatic sequences” like the Morse or Rudin–Shapiro sequences, has been added. The problem of products of the Dirichlet series and some of its specific aspects is examined in depth, with emphasis on the role of the translation 1/2. And the Bohr point of view, which allows to look at a Dirichlet series as at a holomorphic function in several complex variables, is revisited, with some applications like the form of Wiener’s lemma for the Dirichlet series (Hewitt–Williamson’s theorem). The chapter ends with a striking application of this point of view to a density result of Jessen and Bohr. Chapter 5 is a short intermediate chapter establishing the basics of random Dirichlet polynomials through a multidimensional Bernstein inequality and an approach due to Kahane. It will play, technically speaking, an important role in the rest of the book. The tools introduced here remain quite elementary but will turn out to be sufficient for our purposes. Chapter 6 is the longest in the book. It is devoted to the detailed study of new Banach spaces of the Dirichlet series (the Hp -spaces), which extend the initial work of Bohr and turn out to be of basic importance in completeness problems for systems of dilates in the Hilbert space L2 ð0; 1Þ, and seem to open the way to new directions of study, like those of Hankel operators (Helson operators) in infinite dimension. A positive answer to Helson’s conjecture, and related questions, are presented, relying on Harper’s recent finding, admitted here. A complete presentation of a recent, very sharp, version of the Bohnenblust– Hille theorem is also given, using the tools of the previous chapters as well as tools borrowed from number theory, in particular the properties of the function wðx; yÞ, the number of integers  x which are free of prime divisors [y. Chapter 7 gives a complete proof of the universality theorems of Voronin (zeta function) and Bagchi (L-functions), and needs first a reminder of some properties of those functions in the critical strip. This complete proof is long and involved, but some essential tools (like the Birkhoff–Oxtoby ergodic theorem) have already been introduced in the previous chapters. New, important, tools are an extended version of Carlson’s identity seen in Chap. 6, and Hilbertian (Bergman) spaces of analytic functions. Those two results have the advantage of showing the pivotal role of zeta and L-functions in analysis and function theory, in the wide sense, and more or less independently of the Riemann hypothesis. Chapter 8 is a new addition in this second edition. It is devoted to the study of composition operators C u on the Hardy space H2 and their complete characterization by Gordon and Hedenmalm. Some recent works on the membership of Cu in Schatten classes, and to the decay of its singular values, are also

Preface

ix

presented. Finally, using the recent breakthrough of A. Harper on, among others, the L1 -norm of the “Dirichlet kernel” DN ðtÞ ¼

1 X

nit ;

n¼1

we also touch the Hp case when 0\p\2, a case which is not yet completely elucidated. Each of the eight chapters is continued by quite a few exercises, of reasonable difficulty for whoever has read the corresponding chapter. We hope that they can bring additional information and be useful to the reader. Lille, France

Hervé Queffelec Martine Queffelec

Acknowledgments

The idea of writing this book on diophantine approximation and Dirichlet series had been for some time in our minds, under a rather vague form. But the “passage to the act” followed rather quickly a long stay and a complete course on those topics in the Harish-Chandra Institute of Allahabad in January and February 2011. This course was at the invitation of Surya Ramana, Reader in this Institute and a specialist in Number Theory. We take here the opportunity of deeply thanking him for the numerous mathematical discussions we had, as well as for his kindness and efficiency during our stay, and for his patience before the successive delays in the polishing of the final aspects of the book. And the idea of writing a second edition, and his generous advice then, is also due to him. O. Ramaré showed us some significant simplifications of various proofs; his help with the technical aspects of typing, and following the editorial rules of the collection, was invaluable. B. Calado read most of a preliminary version of this book and detected several misprints and errors. K. Seip and O. Brevig had a careful look at Chap. 8, and suggested quite a few improvements. And for the few beautiful (and quite helpful for the reader) pictures, we are indebted to Sumaya Saad-Eddin. All of them are warmly thanked for their patience and expertise.

xi

Contents

1 A Review of Commutative Harmonic Analysis . . . . . . . . . . . . . 1.1 The Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Locally Compact Abelian Groups . . . . . . . . . . . . . 1.1.2 Existence and Properties of the Haar Measure . . . . 1.2 The Dual Group and the Fourier Transform . . . . . . . . . . . . 1.2.1 Characters and the Algebra L1 ðGÞ . . . . . . . . . . . . . 1.2.2 Topology on the Dual Group . . . . . . . . . . . . . . . . 1.2.3 Examples and Basic Facts . . . . . . . . . . . . . . . . . . 1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems . . . . 1.3.1 An Abstract Theorem . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Applications to Harmonic Analysis . . . . . . . . . . . . 1.4 The Inversion and Plancherel Theorems . . . . . . . . . . . . . . . 1.4.1 The Inversion Theorem . . . . . . . . . . . . . . . . . . . . 1.4.2 The Plancherel Theorem . . . . . . . . . . . . . . . . . . . . 1.4.3 A Description of the Wiener Algebra . . . . . . . . . . 1.4.4 A Basic Hilbert-Type Inequality . . . . . . . . . . . . . . 1.5 Pontryagin’s Duality Theorem and Applications . . . . . . . . . 1.5.1 The Pontryagin Theorem . . . . . . . . . . . . . . . . . . . 1.5.2 Topological Applications of Pontryagin’s Theorem 1.6 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Uncertainty Principle on the Real Line . . . . . . 1.6.2 The Uncertainty Principle on Finite Groups . . . . . . 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 4 4 5 7 9 9 11 13 13 17 18 18 21 21 23 27 27 30 34 36

2 Ergodic Theory and Kronecker’s Theorems . . 2.1 Elements of Ergodic Theory . . . . . . . . . . 2.1.1 Basic Notions in Ergodic Theory 2.1.2 Ergodic Theorems . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

37 37 37 38

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

xiii

xiv

Contents

2.2

The Kronecker Theorem . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Statement and Proof of the Main Theorem . . . 2.2.3 A Useful Formulation of Kronecker’s Theorem 2.3 Distribution Problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Distribution in Td . . . . . . . . . . . . . . . . . . . . . 2.3.2 Powers of an Algebraic Number . . . . . . . . . . . 2.4 Towards Infinite Dimension . . . . . . . . . . . . . . . . . . . . 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

45 45 47 50 51 51 54 60 64 66

3 Diophantine Approximation . . . . . . . . . . . . . . . . . . . . . 3.1 One-Dimensional Diophantine Approximation . . . . 3.1.1 Historical Survey . . . . . . . . . . . . . . . . . . . 3.1.2 How to Find the Best Approximations? . . . 3.1.3 Classification of Numbers . . . . . . . . . . . . . 3.1.4 First Arithmetical Results . . . . . . . . . . . . . 3.2 The Gauss Ergodic System . . . . . . . . . . . . . . . . . . 3.3 Back to Transcendence . . . . . . . . . . . . . . . . . . . . . 3.3.1 Metric Results . . . . . . . . . . . . . . . . . . . . . 3.3.2 Simultaneous Diophantine Approximations 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

69 69 69 71 73 76 79 85 85 87 90 94

4 General Properties of Dirichlet Series . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convergence Abscissas . . . . . . . . . . . . . . . . . 4.2.1 The Bohr–Cahen Formulas . . . . . . . . 4.2.2 The Perron–Landau Formula . . . . . . . 4.2.3 The Holomorphy Abscissa . . . . . . . . 4.2.4 A Class of Examples . . . . . . . . . . . . 4.2.5 A More Intricate Dirichlet Series . . . . 4.2.6 Automatic Dirichlet Series . . . . . . . . 4.3 Products of Dirichlet Series . . . . . . . . . . . . . . 4.4 Bohr’s Abscissa via Kronecker’s Theorem . . . 4.4.1 Bohr’s Point of View . . . . . . . . . . . . 4.4.2 The Bohr Inequalities . . . . . . . . . . . . 4.4.3 A Wiener Lemma for Dirichlet Series 4.5 A Theorem of Bohr and Jessen on Zeta . . . . . 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

95 95 97 97 98 99 101 106 110 113 116 116 117 119 121 124 126

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

5 Probabilistic Methods for Dirichlet Series . . . . . . . . . . . . . . . . . . . . . 129 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 A Multidimensional Bernstein Inequality . . . . . . . . . . . . . . . . . . 129

Contents

xv

5.3

. . . 132 . . . 132

Random Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Maximal Functions in Probability . . . . . . . . . . . . . . 5.3.2 The Sub-Gaussian Aspect of Rademacher-Type Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 The Kahane Bound for Random Trigonometric Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Random Dirichlet Polynomials . . . . . . . . . . . . . . . . 5.4 The Proof of Bohnenblust–Hille’s Theorem . . . . . . . . . . . . . 5.4.1 An Elementary Version . . . . . . . . . . . . . . . . . . . . . 5.4.2 A Sharp Version of the Bohnenblust–Hille Theorem 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Hardy Spaces of Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Origin of the Spaces H1 and H2 . . . . . . . . . . 6.1.2 A Basic Property of H1 . . . . . . . . . . . . . . . . . . . 6.2 The Banach Space H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Banach Algebra Structure of H1 . . . . . . . . . . 6.2.2 Behaviour of Partial Sums . . . . . . . . . . . . . . . . . . 6.2.3 Some Applications of the Control of Partial Sums . 6.3 Additional Properties of H1 . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 An Improved Montel Principle . . . . . . . . . . . . . . . 6.3.2 Interpolating Sequences of H1 . . . . . . . . . . . . . . . 6.4 The Hilbert Space H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Definition and Utility . . . . . . . . . . . . . . . . . . . . . . 6.4.2 The Embedding Theorem . . . . . . . . . . . . . . . . . . . 6.4.3 Multipliers of H2 . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Banach Spaces Hp . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 A Basic Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Definition of Hardy-Dirichlet Spaces . . . . . . . . . . . 6.5.3 A Detour Through Harmonic Analysis . . . . . . . . . 6.5.4 Helson Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Harper’s Breakthrough and Two Consequences . . . 6.6 A Sharp Sidon Constant . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Reformulation of Bohr’s Question in Terms of H1 6.6.2 Symmetric Multilinear Forms . . . . . . . . . . . . . . . . 6.6.3 The Claimed Sharp Upper Bound . . . . . . . . . . . . . 6.6.4 A Refined Inclusion . . . . . . . . . . . . . . . . . . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 133 . . . . . . .

. . . . . . .

. . . . . . .

134 135 137 137 139 141 143

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 145 146 147 147 149 151 153 154 155 158 158 160 161 166 166 167 168 171 176 178 178 180 188 192 193 195

xvi

Contents

7 Voronin-Type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 A Reminder About Zeta and L-Functions. . . . . . 7.1.2 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Hilbertian Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 A Hilbertian Density Criterion . . . . . . . . . . . . . 7.2.2 A Density Result in Bergman Spaces . . . . . . . . 7.3 Joint Universality of the Sequence ð‚N Þ . . . . . . . . . . . . . 7.4 A Generalized and Uniform Carlson Formula . . . . . . . . . 7.4.1 Estimates on the Gamma Function . . . . . . . . . . 7.4.2 The Carlson Formula . . . . . . . . . . . . . . . . . . . . 7.5 Joint Universality of the Singleton ‚ ¼ ðLðs; vj ÞÞ . . . . . . 7.5.1 Notations and the Idea of Proof of Theorem 1.2 7.5.2 Details of Proof . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

197 197 197 203 205 205 207 214 218 218 221 226 226 228 233 235

8 Composition Operators on the Space H2 of Dirichlet Series . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Arithmetic Theorem . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Definition and Uniform Convergence . . . . . . . . 8.4.2 Mapping Properties . . . . . . . . . . . . . . . . . . . . . 8.4.3 Twisting and Composition . . . . . . . . . . . . . . . . 8.4.4 Twisting and Probability . . . . . . . . . . . . . . . . . . 8.4.5 Twisting and Topology . . . . . . . . . . . . . . . . . . . 8.5 Integral Representation and Embedding . . . . . . . . . . . . . 8.5.1 Integral Representation . . . . . . . . . . . . . . . . . . . 8.5.2 Embedding Results for H2 . . . . . . . . . . . . . . . . 8.6 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Proof of Necessity in Theorem 8.6.1 . . . . . . . . . 8.6.2 Proof of Sufficiency in Theorem 8.6.1 . . . . . . . . 8.7 Compact Operators and Approximation Numbers . . . . . . 8.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 First Properties of Approximation Numbers . . . . 8.7.3 The Multiplicative Inequalities of H. Weyl . . . . 8.8 A Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 The Case of Hp -spaces . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.1 Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.2 Failure of Embedding for 0\p\2 . . . . . . . . . . 8.10.3 Positive Results . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 239 240 245 245 248 250 251 252 254 254 255 257 257 259 263 263 265 266 268 271 272 272 273 275

Contents

xvii

8.11 A Few Updates and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

About the Authors

Hervé Queffelec is Emeritus Professor at the Department of Mathematics, University of Lille, France. Earlier, he served as Full Professor at Lille 1 University, France (1992–2011) and Assistant Professor at Paris-Sud University, Orsay, France (1968–1992). He completed his Ph.D. in Mathematics in 1971 and received the habilitation à diriger des recherches (HDR) in 1985 from Paris-Sud University, France. His research interests include analytic theory of Dirichlet series, Banach spaces of analytic functions, and Carleson measures. Five students have earned their Ph.D. degrees under the Prof. Queffelec’s supervision during his academic career. He has delivered talks based on his research in renowned universities and institutes around the world, including Harish-Chandra Research Institute, India; Tsinghua University, China; University of Crete, Greece; Euler International Mathematical Institute, Russia; and Centre de Recerca Mathemàtica, Spain. He has published his research papers in several journals of repute. Martine Queffelec is Emeritus Assistant Professor at the Department of Mathematics, University of Lille, France. Earlier, she was Assistant Professor at Lille 1 University, France, (1993–2011), and Assistant Professor at Université Sorbonne Paris Nord, France (1970–1993). She received the habilitation à diriger des recherches (HDR) in 1984 from Université Sorbonne Paris Nord, France. Her areas of research include dynamical systems and ergodic theory, Diophantine approximation and number theory, and harmonic analysis of measures. She has been an active participant of international conferences and has delivered talks in institutes around the world, including Instituto Nacional de Matemática Pura e Aplicada, Brazil; Tsinghua University, China; Research Institute for Mathematical Sciences, Kyoto University, Japan; and Harish-Chandra Research Institute, India. She has three books to her name, including Substitution Dynamical Systems: Spectral Analysis (Springer). Her research papers are published in several journals of repute.

xix

Chapter 1

A Review of Commutative Harmonic Analysis

1.1 The Haar Measure 1.1.1 Locally Compact Abelian Groups This chapter might be skipped at first reading. But we have the feeling that a minimal knowledge of basic facts in harmonic analysis is necessary to understand certain aspects of the analytic theory of Dirichlet series, especially those connected with almost-periodicity, ergodic theory, the Bohr point of view to be developed later, and also universality problems. Therefore, in this introductory chapter, we begin with reminding several basic results of commutative harmonic analysis. Those results, although standard by now, are not so easy to prove, and deserve a careful treatment. Let G be an additive abelian group equipped with a Hausdorff topology τ , which is compatible with the group structure. This means that the operations of the group (addition and inverse) are continuous for that topology. We then say that G is a topological group. Throughout that book, the topology τ will be locally compact, and G will be called a locally compact, abelian group (in short, an LCA group). In most cases, G will indeed be compact. A basic example is that of the compact multiplicative group T of unimodular complex numbers, that is the unit circle of the complex plane C. This particular group plays a fundamental role in the theory. For a ∈ G, we will denote by Ta the operator of translation by a (a homeomorphism of G, also acting on functions), namely Ta x = x + a, Ta f (x) = f (x + a).

(1.1.1)

A simple and useful result is the following: Proposition 1.1.1 Let G be a topological group and H a subgroup of G. If H has non-empty interior, H is open and closed in G.

© Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_1

1

2

1 A Review of Commutative Harmonic Analysis

Proof Let a be interior to H and V a neighbourhood of 0 with a + V ⊂ H . For any b ∈ H , we have b + V = (b − a) + (a + V ) ⊂ H , showing that H is open, as well as its cosets x + H . Now, we can write G as a disjoint union G = H  E, where E is a union of cosets mod H and is open. So that H = G\E is closed. 

1.1.2 Existence and Properties of the Haar Measure A basic fact, expressing the strong ties between the topology and the group structure, is the following theorem: Theorem 1.1.2 A locally compact abelian group G always possesses a non-zero, positive and regular Borel measure m which is translation invariant, i.e. 

 f (x)dm(x) = G

f (Ta x)dm(x) ∀ f ∈ L 1 (G, m), ∀a ∈ G.

(1.1.2)

G

This measure (also written dx) is unique up to multiplication by a positive scalar and we write L 1 (G) instead of L 1 (G, m). A very simple proof can be found in [1] (pp. 570–571) in the case of a compact, metrizable, group, abelian or not. A clear and modern proof can be found for the general (abelian, but not necessarily metrizable) case in [2], Chap. 9. This generality will sometimes be needed, as shown by the forthcoming examples. The measure m is called the Haar measure of G. Three important properties of m are the following: Proposition 1.1.3 The Haar measure verifies: m (V ) > 0 for each open non-void set V ⊂ G.

(1.1.3)

m(−B) = m (B) for all Borel subsets of G. m (G) < ∞ ⇐⇒ G is compact.

(1.1.4) (1.1.5)

Indeed, suppose that m(V ) = 0. Let K ⊂ G be a compact set. This set can be covered by finitely many translates of V , and therefore, m(K ) = 0. But since m is regular, we have m(G) = sup K ⊂G m(K ) so that m(G) = 0, which is absurd. Now, the measure m˜ defined by m(B) ˜ = m(−B) is translation invariant, therefore m˜ = cm where c is a scalar. Let then V be a compact and symmetric neighbourhood of 0, so that by (1.1.3) we have 0 < m(V ) < ∞. The equation cm(V ) = m(V ), therefore, implies c = 1 and m˜ = m. Suppose that G is not compact, and observe that (just take x outside the compact set K − L): If K , L ⊂ G are compact, there exists x ∈ G : (x + L) ∩ K = ∅.

(1.1.6)

1.1 The Haar Measure

3

Now, let V be a compact neighbourhood of 0, so that m(V ) > 0 by (1.1.3). Using (1.1.6), we can inductively find a sequence (xn ) ⊂ G such that the translated sets x j + V are disjoint. Therefore, for any n ≥ 1: m(G) ≥ m

n 



(x j + V ) =

j=1

n 

m(x j + V ) = n × m(V ),

j=1

and this shows that m(G) = ∞. If G is compact, m is clearly finite and we always normalize it to have m(G) = 1, i.e. m is a probability measure.  In the general case, let M(G) be the set of regular, complex Borel measures on G, normed with the total variation of measures. By the Riesz representation theorem, M(G) can be isometrically identified with the dual of the Banach space C0 (G) of continuous functions f : G → C which tend to zero at infinity, namely: ∀ε > 0, ∃K ⊂ G, K compact; x ∈ / K =⇒ | f (x)| ≤ ε. The convolution λ ∗ μ of λ and μ in M(G) is the element σ of M(G) defined on Borel sets E by:  λ(E − x)dμ(x) = μ(E − x)dλ(x), G G   equivalently f dσ = f (x + y)dλ(x)dμ(y). 

σ(E) =

One can define an involution μ → μ˜ on M(G) by the formula: μ(E) ˜ = μ(−E). Once equipped with the variation-norm, convolution and involution, M(G) is a commutative, unital (the unit being the Dirac measure δ0 at the origin), stellar (meaning that μ ˜ = μ) Banach algebra. But this is not a C ∗ -algebra: the equation 2 μ ∗ μ ˜ = μ does not hold in general. The Banach space L 1 (G) = L 1 (G, m) is a closed ideal of M(G), the ideal of measures which are absolutely continuous with respect to m. It is itself a commutative Banach algebra once equipped with the convolution f ∗ g as multiplication:  f (x − y)g(y)dm(y) = (g ∗ f )(x),

( f ∗ g)(x) = G

for almost every x ∈ G. We have  f ∗ g1 ≤  f 1 g1 and the algebra L 1 (G) is unital if and only if G is compact. This is an involutive algebra with the induced involution defined by f˜(x) = f (−x), i.e. we have  f˜1 =  f 1 . But this is not a C ∗ -algebra either: the equation  f ∗ f˜1 =  f 21 does not hold in general (see the exercises). Another important property of L 1 (G) is the following general fact:

4

1 A Review of Commutative Harmonic Analysis

Theorem 1.1.4 Let f ∈ L p (G), 1 ≤ p < ∞. Then, the mapping a → Ta f : G → L p (G) is (uniformly) continuous. Proof The result holds, by uniform continuity, for h ∈ C00 (G), the space of functions: G → C which are continuous and compactly supported. This space is dense in L p (G) since p < ∞. And by translation invariance of m, we clearly have: Ta f − f  p ≤ 2h − f  p + Ta h − h p , which gives the general result, since Ta f − Tb f  p = Ta−b f − f  p .



We will see that the spectrum of L 1 (G) can be identified, whereas a complete description of the spectrum of M(G) is difficult to obtain, and to work with [3]. To that effect, we first have to define the dual of an LCA group.

1.2 The Dual Group and the Fourier Transform 1.2.1 Characters and the Algebra L 1 (G)  or  of the LCA group G is the group of all continuous morphisms The dual group G γ : G → T, i.e. |γ(x)| = 1; γ(x + y) = γ(x)γ(y) ∀x, y ∈ G. The elements of  are called the (continuous, or strong) characters of G. Sometimes, we will consider all the characters, continuous or not, on G. They are called the weak characters. The set , equipped with the natural multiplication of characters, is itself an abelian group (for multiplication) whose zero element is the character identical to one. And γ −1 = γ for each γ ∈ . This group appears naturally for the following reason: Theorem 1.2.1 The spectrum L of the Banach algebra L 1 (G) can be naturally identified with , in the following sense: (1) Each γ ∈  defines h γ ∈ L by the formula  hγ ( f ) =

γ(−x) f (x)dm(x). G

(2) Each element h ∈ L is of the form h = h γ .  Generally, G γ(−x) f (x)dm(x) is denoted by  f (γ) and is called the Fourier transform of f at γ. If G is compact and moreover f ∈ L 2 (G), we see that  f (γ) =  f, γ, the scalar product of f and γ. In view of Theorem 1.2.1 (see [4],

1.2 The Dual Group and the Fourier Transform

5

p. 7 for a detailed proof), we will naturally equip  with the Gelfand topology of L, which is the weak-star topology inherited from the dual space Y of L 1 (G). This makes  a compact Hausdorff space if L is unital, which happens if and only if G is discrete, and a locally compact Hausdorff space in the general case (since L ∪ {0} is weak-star-closed and therefore weak-star-compact in the unit ball of Y ). But this topology is fairly abstract and difficult to describe, and we will see later a more concrete and tractable definition. It is first useful to study in detail this Fourier transform, whose main properties are listed in the simple, following theorem, and with obvious notations. Theorem 1.2.2 The Fourier transform on L 1 (G) satisfies (1) (2) (3) (4) (5)

f ∈ C0 () and   f ∞ ≤  f 1 If f ∈ L 1 (G),  If γ1 = γ2 , there exists f ∈ L 1 (G) ∩ L 2 (G);  f (γ1 ) =  f (γ2 ) f (γ) = 0 For any γ ∈ , there exists f ∈ L 1 (G) ∩ L 2 (G);  If f , g ∈ L 1 (G),  f ∗g =  f g   f and  γ0 f (γ) =  f (γγ0 ). f ∗ γ = f (γ)γ; Ta f = γ(a) 

Let us denote by A() the subspace of C0 () formed by functions of the form g (γ) =  f (γ) for some f ∈ L 1 (G). This set is called the Wiener algebra of . We have the following corollary of Theorem 1.2.2: Corollary 1.2.3 The space A() is a dense, self-adjoint, subalgebra of C0 (), stable under translation and multiplication by a character. Proof Using the items of Theorem 1.2.2, we see that A() is a subalgebra, which separates points of  and has no common zeros. If g =  f ∈ A(), so does g =  f˜ as we easily see. Therefore, the complex Stone–Weierstrass theorem for locally compact  spaces applies and A() is uniformly dense in C0 ().

1.2.2 Topology on the Dual Group Here is now an alternative description of the topology on  ([4], pp. 10–11). One interest of this description is that it shows the following: the set , which is so far an abelian group and a locally compact Hausdorff space, is indeed a locally compact abelian group. Theorem 1.2.4 The natural topology on  is that of uniform convergence on compact subsets of G. More precisely, K, C being compact subsets of G and , respectively, and r a positive number, we have (1) The function (x, γ) → γ(x) is continuous on G × . (2) Let N (K , r ) = {γ ∈ ; |1 − γ(x)| < r for all x ∈ K }. Then, N (K , r ) is an open subset of .

6

1 A Review of Commutative Harmonic Analysis

(3) The family of all sets N (K , r ) and their translates is a base for the topology of . (4) Let M(C, r ) = {x ∈ G; |1 − γ(x)| < r for all γ ∈ C}. Then, M(C, r ) is an open subset of G. (5)  itself is a locally compact abelian group. Proof (1) Let (x0 , γ0 ) ∈ G × . By Theorem 1.2.2, there is f ∈ L 1 (G) such that  f (γ0 ) = 0, and we can write, near (x0 , γ0 ): γ(x) =

 Tx f (γ) .  f (γ)

The denominator is continuous at (x0 , γ0 ) by Theorem 1.2.2. The numerator as well, since setting g = Tx0 f , we see that      | Tx f (γ) − T x0 f (γ0 )| ≤ | Tx f (γ) − Tx0 f (γ)| + | Tx0 f (γ) − Tx0 f (γ0 )| ≤ Tx f − Tx0 f 1 + | g (γ) −  g (γ0 )|, and the right-hand side tends to 0 as (x, γ) → (x0 , γ0 ), by Theorems 1.1.4 and 1.2.2. (2) Now, fix γ0 ∈ N (K , r ). For each x ∈ K , there are open neighbourhoods Vx and Wx of x and γ0 respectively such that y ∈ Vx and γ ∈ Wx =⇒ |γ(y) − 1| < r. p

Let Vx1 , . . . , Vx p be a finite covering of K and W = ∩ j=1 Wx j . The set W is a neighbourhood of γ0 and W ⊂ N (K , r ), so that N (K , r ) is open in . (3) Conversely, let V be a neighbourhood of γ0 . We may assume that γ0 = 1. By definition of the Gelfand topology on , there are functions f 1 , . . . , f n ∈ L 1 (G) and ε > 0 such that n {γ; |  f j (γ) −  f j (1)| < ε} ⊂ V. (1.2.1) W = j=1

By density, we may assume that f 1 , . . . , f n ∈ C00 (G), so that they vanish outside a compact set K ⊂ G. If r < ε/ max j  f j 1 , one easily checks that N (K , r ) − W ⊂ V , since     |1 − γ(−x) f j (x)|d x = |1 − γ(x) f j (x)|d x < ε. | f j (γ) − f j (1)| ≤ K

K

(4) The same proof applies to M(C, r ), with a significant difference: the sets M(C, r ) and their translates will turn out to be a base for the topology of G. But so far we are unable to establish that fact, which will be proved and used later, and have to content ourselves with the sets N (K , r ).

1.2 The Dual Group and the Fourier Transform

7

(5) The obvious inequality |1 − δ1 (x)δ2 (x)| ≤ |1 − δ1 (x)| + |1 − δ2 (x)|, ∀δ1 , δ2 ∈ , ∀x ∈ G shows that [γ1 × N (K , r/2)][γ2 × N (K , r/2)] ⊂ γ1 γ2 × N (K , r ). This and the previous description of the topology of  shows that the map (γ1 , γ2 ) → γ1 γ2 is continuous, so that  is a LCA group. 

1.2.3 Examples and Basic Facts Let us now list, sometimes without proof, some basic examples and facts about Haar measures and dual groups. 1.2.3.1 The dual of a compact group is a discrete one, and the dual of a discrete group is a compact one.

d = Zd and if γ = (n 1 , . . . , n d ) ∈ Zd , z = (z 1 , . . . , z d ) ∈ Td we have 1.2.3.2  T n γ(z) = dj=1 z j j . The Haar measure m of Td acts on continuous functions by the formula  1 1  1  f dm = ... f (e2iπt1 , . . . , e2iπtd )dt1 . . . dtd . Td

0

0

0

d = Td . This last fact will later appear as a consequence of the PonSimilarly, Z tryagin duality theorem. More generally, if G 1 , . . . , G d are locally compact abelian groups with Haar measures m 1 , . . . , m d and dual groups 1 , . . . , d , the product group G = G 1 × · · · × G d has the Haar measure m = m 1 ⊗ · · · ⊗ m d and its dual group is  = 1 × · · · × d . ∞ = Z(∞) where the LHS is the product of countably many copies of T and 1.2.3.3 T the RHS is the set of all sequences  ν =n j (n 1 , . . . , n d , . . .) of integers which vanish for d large enough, with γ(z) = ∞ j=1 z j , all but a finite number of the factors being equal to 1. The Haar measure of T∞ is the tensor product of countably many copies of the Haar measure of T. This fact has an obvious generalization to the countable product of compact abelian groups, as in Example 2.

d = Rd and if γ = (t1 , . . . , td ) ∈ Rd , x = (x1 , . . . , xd ) ∈ Rd we have 1.2.3.4 R

d γ(x) = ei j=1 t j x j . The Haar measure of Rd is simply the Lebesgue measure on Rd . Those facts follow from the general remark of Example 2. 1.2.3.5 Let G be a compact abelian group with dual . Then, we have the equivalence: G metrizable ⇐⇒  countable.

(1.2.2)

We use the following fact: if X is a topological compact space and C(X ) the space of continuous functions f : X → C equipped with the norm  f ∞ = supt∈X | f (t)|, we have the equivalence:

8

1 A Review of Commutative Harmonic Analysis

X metrizable ⇐⇒ C(X ) separable.

(1.2.3)

Indeed, if the topology of X is defined by a metric d, let (xn ) be a dense sequence of X , and ϕn (x) = d(x, xn ). The algebra generated by the ϕn is separable, and dense in C(X ) by the Stone–Weierstrass theorem. Conversely, if ( f n ) is a dense subset of C(X ), the distance d defined by d(x, y) =

∞ 

2−n

n=1

| f n (x) − f n (y)| 1 + | f n (x) − f n (y)|

is easily seen to define the topology of X . To prove (1.2.2), we observe that  ⊂ C(G) and that, if γ, γ  ∈  are distinct, we have by orthogonality: γ − γ  ∞ ≥ γ − γ  2 =

√

2.

(1.2.4)

Now, if  is countable, the set P of trigonometric polynomials is separable, and dense in C(G) by Theorem 1.3.4 to come. Therefore, G is metrizable by (1.2.3). Conversely, if G is metrizable, C(G) is separable, and then  has to be countable in view of (1.2.4). This ends the proof of (1.2.2). 1.2.3.6 Let R be equipped with the Haar measure dx, the usual Lebesgue measure. Its dual  can be identified to R, but then the Haar measure corresponding to the forthcoming inversion Theorem 1.4.1 is d x/2π. Indeed, if f (t) = e−|t| , one easily computes  f (x) = 2/(1 + x 2 ) and the change of variable x = tan t shows that 1 2π



4 | f (x)|2 d x = π R



π 2

0

 cos2 tdt = 1 =

R

| f (t)|2 dt.

1.2.3.7 If G = {x1 , . . . , x N } is a finite abelian group with dual  = {γ1 , . . . , γ N } (isoN δxi , morphic to G), and if we equip G with the normalized Haar measure m = N1 i=1 the Haar measure on  corresponding to the inversion theorem is the non-normalized

measure μ = Nj=1 δγ j as is easily checked. This corresponds to the fact that the matrix ( √1N γ j (xi ))(i, j) is unitary. This example is very important for Dirichlet characters. 1.2.3.8 As an important specialization of Example 3, we have the following: let G be the Cantor group, i.e. the compact abelian and metrizable group {−1, 1}N of all choices of signs ω = (εn )n≥1 with εn = ±1 and co-ordinatewise multiplication, equipped with its normalized Haar measure m. Its dual group (discrete and countable) is called the Walsh group and can be described as the group of words w A indexed by the finite subsets of N∗ := {1, 2, . . .} defined by w A (ω) =

 n∈A

εn (ω), w∅ (ω) = 1.

1.2 The Dual Group and the Fourier Transform

9

The coordinate functions εn are independent random variables on the probability space (G, m) and are sometimes called the Rademacher, or centered Bernoulli, variables. They will play a very important role in the study of random polynomials and random Dirichlet series.

1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems 1.3.1 An Abstract Theorem The structure of stellar, Banach algebra of M(G) is interesting for us with a view to the following fundamental theorem. Let A denote a commutative, stellar, unital Banach algebra with unit e, with dual space A∗ (in the sense of Banach spaces), involution x → x˜ and spectrum M. We recall that M is the set of non-zero homomorphisms ϕ : A → C, which are automatically continuous with norm 1. This is a compact Hausdorff space with the usual Gelfand topology, namely the weak-star topology x (γ) = γ(x) the Gelfand transform of x ∈ A at induced by A∗ on M. We denote by  γ ∈ M, and by r (x) :=  x ∞ the spectral radius of x ∈ A. We then have the: Theorem 1.3.1 (Bochner–Weil–Raikov) Let L be a positive linear form on A, namely L (x x) ˜ ≥ 0 for all x ∈ A. Then, we have (1) L is continuous. (2) |L(x)| ≤ L(e)r (x) and |L (x x) ˜ | ≤ L(e)r (x)2 for all x ∈ A. (3) There is a positive measure μ on M such that  L(x) =

M

 x (γ)dμ(γ), ∀x ∈ A.

(4) If L(x x) ˜ = 0, there exists χ ∈ M such that χ(x) = 0. Proof (1) First note that e˜ = e since e˜ is also a unit for A. Now recall that, for t real and |t| ≤ 1, it holds: √

1−t =

∞ 

an t n with an real and

n=0

∞ 

|an | < ∞.

n=0

So that, if x ∈ A and x ≤ 1, we can write e − x x˜ = y 2 with y = y˜ =

∞ 

an (x x) ˜ n.

n=0

This proves that L(e − x x) ˜ = L(y y˜ ) ≥ 0 and that L(x x) ˜ ≤ L(e). Moreover, the assumptions imply that the map (x, y) → L(x y˜ ) is a positive, Hermitian, form on

10

1 A Review of Commutative Harmonic Analysis

A, therefore we have the Cauchy–Schwarz inequality: ˜ y˜ ). |L(x y˜ )|2 ≤ L(x x)L(y Taking y = e, we get, for x ≤ 1, the following: ˜ ≤ [L(e)]2 . |L(x)|2 ≤ L(e)L(x x)

(1.3.1)

(2) Using (1.3.1) and then iterating, we get n

˜ 2 |L(x)| ≤ L(e)1/2+1/4+···+1/2 [L(x x) n

n

≤ L(e)1/2+1/4+···+1/2 L1/2 (x x) ˜ 2

n−1

n−1

]1/2

n

n

1/2 .

Recall that, according to the spectral radius theorem, r (x) is given by r (x) = lim x n 1/n . n→∞

(1.3.2)

So that, letting n tend to infinity in the above, we get the first claimed inequality |L(x)| ≤ L(e)r (x x) ˜ 1/2 ≤ L(e)r (x). ˜ and χ(x x) ˜ = χ(x)ψ(x), Indeed, if χ ∈ M, so does ψ defined by ψ(x) = χ(x), ˜ so that r (x x) ˜ ≤ r (x)2 . The second inequality follows by changing x into x x.  the subspace of C(M) formed by Gelfand transforms of elements of (3) Let A  by the formula S( A. Define a linear form S on A x ) = L(x). The preceding shows that S is well-defined and that |S( x )| ≤ L(e) x ∞ .  and S ≤ L(e). The Hahn–Banach extension Therefore, S is continuous on A theorem and the Riesz representation theorem now show that there exists a regular, complex measure μ on M, with μ ≤ L(e), such that:  L(x) = S( x) =

M

 x (γ)dμ(γ).

 In particular, L(e) = S(1) = M dμ(γ) ≥ μ, so that μ is positive with norm L(e). (4) If L(x x) ˜ = 0, item (1) shows that r (x) =  x ∞ = 0, which ends the proof. 

1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems

11

1.3.2 Applications to Harmonic Analysis An important consequence of Theorem 1.3.1 is: Theorem 1.3.2 The Fourier transform f →  f : L 1 (G) → A() is injective. Proof Consider the unital subalgebra A = L 1 (G) + Cδ0 of M(G). Fix a function ϕ in C00 (G), the set of continuous, compactly supported functions G → C. Then, define a linear form L = L ϕ on that algebra by the formula: L(σ) = (ϕ˜ ∗ ϕ ∗ σ) (0), that is, if σ = f dm + cδ0 ∈ A: L(σ) = (ϕ˜ ∗ ϕ ∗ f ) (0) + c (ϕ˜ ∗ ϕ) (0). This linear form is positive, since one easily sees that: L(σ ∗ σ) ˜ = σ ∗ ϕ22 . (Observe in passing that, by Cauchy–Schwarz, Fubini and the translation invariance of m, one has for σ ∈ M(G) : ϕ ∗ σ ∈ L 2 (G), with moreover ϕ ∗ σ2 ≤ ϕ2 σ). Now, let f ∈ L 1 (G), f = 0. Choose ϕ ∈ C00 (G) such that  ( f ∗ ϕ)(0) =

ϕ(−x) f (x)d x = 0.

(1.3.3)

G

This implies that L( f ∗ f˜) =  f ∗ ϕ22 = 0 since f ∗ ϕ is continuous, does not vanish at 0 by (1.3.3), and since the Haar measure charges all non-void open sets by (1.1.3). Therefore, by Theorem 1.3.1, there is a character h of A such that h( f ) = 0. But the characters of A are of the form: f (γ) + c, for soe γ ∈ . h( f dm + cδ0 ) =  Taking c = 0 here, we obtain h( f ) =  f (γ) = 0, which gives the result.



In functional analysis, the dual of a normed space has many elements thanks to the Hahn–Banach theorem. It turns out that the dual of a locally compact abelian group G has many elements as well. Namely, as a consequence of Theorem 1.3.2, we have the Peter–Weyl theorem in the abelian case: Theorem 1.3.3 (Peter–Weyl theorem) The dual  of any LCA group G separates the points of G, namely: I f x = y, there exists γ ∈ ; γ(x) = γ(y).

(1.3.4)

Proof Let x, y ∈ G with x = y. By the Tietze–Urysohn theorem, there exists ϕ ∈ C00 (G) such that ϕ(x) = ϕ(y), that is Tx ϕ(0) = Ty ϕ(0). Applying Theorem 1.3.2, we can find γ ∈  such that  Tx ϕ(γ) =  Ty ϕ(γ), equivalently: ϕ (γ)γ(x) = ϕ (γ)γ(y), so that γ(x) = γ(y).



12

1 A Review of Commutative Harmonic Analysis

Let us indicate some important consequences of the Peter–Weyl theorem. We will denote by P be the algebra of trigonometric polynomials on G, i.e. the vector space of functions generated by . Theorem 1.3.4 If G is a compact abelian group, the set P of trigonometrical polynomials is uniformly dense in the space C(G) of complex, continuous functions on G. Conversely, if  is a subgroup of  separating the points of G, we have  = . Proof The set P is a self-adjoint algebra, since the conjugate of a character, and the products of two of them, is still a character. It separates points of G by the Peter–Weyl Theorem 1.3.3, and contains the constant 1, the zero-character. Therefore, it is dense in C(G) by the Stone–Weierstrass theorem. Now, let Q be the set of trigonometric polynomials generated by , i.e. the vector space generated by . This is a selfadjoint algebra since  is a subgroup, and it separates points of G, therefore is uniformly dense in C(G) by the Stone–Weierstrass theorem again. Now, suppose that γ ∈ \, and let Q ∈ Q. By orthogonality, we have:  γ − Q ∞ ≥ γ − Q 2 = (1 + Q22 )1/2 ≥ 1, which contradicts the uniform density of Q in C(G).



A nice partial consequence of Theorem 1.3.4 is a kind of Hahn–Banach extension theorem for certain subgroups. A more complete description will be given once we have the Pontryagin theorem at our disposal. Corollary 1.3.5 Let H be a subgroup of the LCA group G. Then: (a) Any weak character of H extends to a weak character of G. (b) If H is compact or open, any continuous character on H extends to a continuous character on G. Proof (a) We use a transfinite induction (or Zorn’s lemma) as follows: Let (K , δ) be a maximal pair formed by a subgroup K with H ⊂ K ⊂ G and a weak character δ on K extending γ. If K = G, let x ∈ / K and L be the group generated by K and x. We separate two cases: (1) nx ∈ / K for any non-zero integer n. Let w ∈ T. Then, the formula ε(k + nx) = δ(k)w n gives a well-defined extension of δ to a character ε on L. (2) Otherwise, let p be the smallest positive integer such that px ∈ K and let z = δ( px). We now use the (only) fact that T is a divisible group, namely: ∀z ∈ T, ∀ p ∈ N, ∃w ∈ T; w p = z.

(1.3.5)

We then set ε(k + nx) = δ(k)w n where w is as in (1.3.5) and this still gives a well-defined extension of δ to a character ε on L.

1.3 The Bochner–Weil–Raikov and Peter–Weyl Theorems

13

In all cases, we get a contradiction with maximality, so that we have K = G, as we wish.  be the dual of H and  the subgroup of (b) Suppose first that H is compact. Let H  formed by restrictions to H of characters in . By the Peter–Weyl theorem,  H . If H is open, separates the points of H . By Theorem 1.3.4, we conclude that  = H  and δ be a weak character on G extending γ. This character is continuous at let γ ∈ H the origin, since γ is continuous and H open. Therefore, it is continuous everywhere, i.e. δ ∈ . An extension of that corollary will be given after the Pontryagin duality theorem.  Another fundamental consequence of the Peter-Weyl theorem is: Theorem 1.3.6 If G is compact, the group  is an orthonormal basis of the Hilbert space L 2 (G). Proof We will abbreviate dm(x) in dx. The set  is a normed system in L 2 (G), since if γ ∈ :   γ22 =

|γ(x)|2 d x = G

If γ = 1, let a such that γ(a) = 1 and I =





 G

G

γ(x)d x. We have



γ(x)d x =

I =

d x = 1. G

γ(x + a)d x = G

γ(x)γ(a)d x = γ(a)I, G

and  this implies I = 0. If γ1 , γ2 ∈  and γ1 = γ2 , setting γ = γ1 γ2 = 1, we get G γ1 (x)γ2 (x)d x = G γ(x)d x = 0, showing that  is an orthonormal system. Moreover, the vector space P generated by  is dense in C(G) (by Theorem 1.3.4), itself  dense in L 2 (G), and that ends the proof.

1.4 The Inversion and Plancherel Theorems 1.4.1 The Inversion Theorem Throughout this section, G is a locally compact abelian group,  its dual and C00 (G) the set of continuous functions f : G → C with compact support. It will also be useful to use the following test space: E = L 1 (G) ∩ L 2 (G) ⊃ C00 (G).

(1.4.1)

This space, which is dense in both L 1 (G) and L 2 (G), will play an important role in the rest of this section. We will denote by C the convex cone of functions υ : G → C of the form

14

1 A Review of Commutative Harmonic Analysis

υ=

n 

u j ∗ u˜ j , u 1 , . . . , u n ∈ E.

j=1

Thanks to the obvious identity (γu) ∗ (γυ) = γ(u ∗ υ) for any u, υ ∈ L 1 (G) and γ = γ˜ ∈ , the set C is invariant by multiplication by any γ ∈  and clearly included in L 1 (G). Also note that if υ ∈ C, one has: υ=

n 

u j ∗ u˜ j =⇒  υ (γ) =

j=1

p 

|u j (γ)|2 ≥ 0.

j=1

Since  is a locally compact abelian group, it has a Haar measure, for abstract reasons. Here, we will let this Haar measure appear in a more effective way, through the Fourier transform of elements of C. Theorem 1.4.1 The cone C has the following properties: (1) For any υ ∈ C, there exists a unique positive measure μ = μυ on  such that  υ(x) =



γ(x)dμ(γ) =: Fμ (x), for all x ∈ G.

(1.4.2)

(2) The measures μυ verify the coherence relation dμυ , for all υ, w ∈ C.  υ dμw = w

(1.4.3)

(3) There is a Haar measure dγ on  such that υ ∈ C =⇒  υ ∈ L 1 () and:  γ(x) υ (γ)dγ, for all x ∈ G. (1.4.4) υ(x) = 

Proof 1. First observe the following identity, valid for any f ∈ L 1 (G) and μ ∈ M(), and which is a straightforward consequence of the Fubini theorem:  

 f (γ)dμ(γ) =

 f (x)Fμ (−x)d x.

(1.4.5)

G

Now, if Fμ = 0, we see that μ annihilates all functions in A(), which is dense in C0 () by Corollary 1.2.3 and therefore μ = 0, which gives the uniqueness claimed. As concerns the existence, we may assume that υ = u ∗ u. ˜ Let A = L 1 (G) + Cδ0 and L : A → C defined by L(σ) = (υ ∗ σ)(0). This is a positive linear form since, as we saw in the proof of Theorem 1.3.1: L(σ ∗ σ) ˜ = u ∗ σ22 (≤ u22 σ2 ).

1.4 The Inversion and Plancherel Theorems

15

Therefore, using the Bochner–Weil–Raikov theorem and specializing to the measure σ = f dm, we get the existence of a positive μ ∈ M() such that  (υ ∗ f ) (0) =



 f (γ)dμ(γ), for all f ∈ L 1 (G).

Equivalently, in view of (1.4.5): 





υ(x) f (−x)d x = G

f (x)Fμ (−x)d x = G

f (−x)Fμ (x)d x. G

Since f is arbitrary, it follows that υ = Fμ , as claimed. 2. Let υ, w ∈ C and f ∈ L 1 (G). By associativity and commutativity of convolution on L 1 (G), we have   f (γ) υ (γ)dμw (γ) = [( f ∗ υ) ∗ w] (0) = [( f ∗ w) ∗ υ] (0)    = f (γ) w (γ)dμυ (γ). 

By density of A() in C0 (), Eq. (1.4.3) follows. υ = dμw / w is independent of 3. The idea is the following: (1.4.3) implies that dμυ / v ∈ C. This common value will be a Haar measure on . Let us elaborate: we define a positive linear form T on C00 () by the formula  T (ϕ) =



ϕ dμυ ,  υ

(1.4.6)

where υ ∈ C is > 0 on the support K of ϕ. Such a υ clearly exists. Indeed, for each u = 0 on a neighγ ∈ K , by 3. of Theorem 1.2.2, there exists u = u γ ∈ E such that  p bourhood of γ. Cover K by finitely many Vγ1 , . . . , Vγ p and take υ = j=1 u γ j ∗ u ∼ γj . Thanks to (1.4.3), T (ϕ) does not depend on the choice of υ with  υ > 0 on K , and is linear. Let us further note the identity:  υ) = For any ϕ ∈ C00 () and for any υ ∈ C, T (ϕ

ϕdμυ .

(1.4.7)

Indeed, let w ∈ C with w  > 0 on the support of ϕ and then also on the support of ϕ υ . We get from (1.4.3) that  T (ϕ υ) =

ϕ υ

dμw = w 

 ϕ w

dμυ = w 

 ϕdμυ .

Now, take υ ∈ C, υ= 0. It follows from (1.4.5) that μ = μυ = 0, so there υ ) > 0. This shows that T = 0. Finally, fix exists ϕ ∈ C00 () with  ϕdμυ = T (ϕ v > 0 on K , and let ϕ ∈ C00 (G) with support K and γ0 ∈ . Take υ ∈ C with 

16

1 A Review of Commutative Harmonic Analysis

w = γ0 υ ∈ C. We first note that, τ denoting the translation Tγ0 , we have  μw = τ (μυ ) or

 h(γ)dμw (γ) =

h(γγ0 )dμυ (γ).

(1.4.8)

Indeed, let σ = τ (μυ ). Formula (1.4.5) shows that    w(x) = (γγ0 )(x)dμυ (γ) = γ(x)dσ(γ) = γ(x)dμw (γ), so that σ = μw by the uniqueness of 1. Now, let ψ = Tγ0 ϕ, i.e. ψ(γ) = ϕ(γγ0 ). We see that w (γ) =  υ (γγ0 ) > 0 on the support K γ0 of ψ, so that using (1.4.8):  T (ψ) =

ψ(γ) dμw (γ) = w (γ)



ϕ(γγ0 ) dμw (γ) =  υ (γγ0 )



ϕ(γ) dμυ (γ) = T (ϕ).  υ (γ)

This shows that T = 0 is translation invariant, and so there is a Haar measure dγ on  such that  T (ϕ) = ϕ(γ)dγ for all ϕ ∈ C00 (). (1.4.9) 

Now, if υ ∈ C, it follows from (1.4.7) that   υ (γ)dγ = ϕdμυ . T (ϕ υ ) = ϕ(γ) 

υ (γ)dγ = dμυ . Since μυ is finite, we Since ϕ ∈ C00 () is arbitrary, we obtain  have  υ ∈ L 1 (). And the last equality gives us 

 υ(x) =

γ(x)dμυ (γ) =

γ(x) υ (γ)dγ,

proving (1.4.4).



One first important consequence of the inversion theorem is Corollary 1.4.2 The sets M(C, r ) in Theorem 1.2.2 and their translates form a base for the topology of G. Proof Let V be a neighbourhood of√0 in G and W a compact neighbourhood of 0 such that W − W ⊂ V . Let u = 1W / m(W ) and υ = u ∗ u˜ ∈ E, with support in V . Since υ(0) = 1, the inversion υ (γ)dγ = 1, so there is a compact set  formula gives  υ (γ)dγ > 2/3. With the notations of Theorem 1.2.2, C ⊂  such that we have C  we will show that M(C, 1/3) ⊂ V . Indeed, suppose that x ∈ M(C, 1/3). Then, C  denoting the complement of C in :

1.4 The Inversion and Plancherel Theorems

17

 υ(x) = 1 +



[γ(x) − 1] υ (γ)dγ



≥1−

|γ(x) − 1| υ (γ)dγ − |γ(x) − 1| υ (γ)dγ C   1 2 1  υ (γ)dγ − 2  υ (γ)dγ > 1 − − = 0, ≥1−  3 C 3 3 C C

thus x ∈ V , which ends the proof. This description of the topology of G will be useful in the proof of the Pontryagin duality theorem. 

1.4.2 The Plancherel Theorem A main consequence of the inversion formula is the Theorem 1.4.3 (Plancherel theorem) The Haar measure of  being fixed as in the inversion theorem, there exists an isometric isomorphism  of L 2 (G) onto L 2 () such that  ( f ) =  f for all f ∈ L 1 (G) ∩ L 2 (G). Moreover:  ( f g) = ( f ) ∗ (g) for all f, g ∈ L 2 (G).

(1.4.10)

Proof Let f ∈ E, so that υ = f ∗ f˜ ∈ C with  υ = | f |2 . From Theorem  1.4.1,2 we 1 υ (γ)dγ, that is G | f (x)|2 d x =  |  f (γ)| dγ, know that  υ ∈ L () with υ(0) =   showing that the Fourier transform is an isometry from E to L 2 (). By density of E in L 2 (G), this isometry can be extended into an isometry  : L 2 (G) → L 2 (), which will have closed range V . Let us show that V is dense, by showing that its orthogonal V ⊥ is reduced to {0}: Let u ∈ V ⊥ . If f ∈ E and x ∈ G, we have Tx f ∈ E, f (γ)dγ ∈ M(), we have with the notation of (1.4.5): so that if μ = u(γ)   Fμ (x) =



u(γ)  f (γ)γ(x)dγ =

 

u(γ) Tx f (γ)dγ = 0.

By the uniqueness property of Theorem 1.4.1, we get u(γ)  f (γ) = 0, dγ-almost f = 0 in a everywhere. Now, if γ0 ∈ , by Theorem 1.2.2 there is f ∈ E with  neighbourhood of γ0 . This shows that u(γ) = 0 a.e. and that u = 0, so that V = L 2 (). As all linear isometries between Hilbert spaces,  preserves the inner product: 

 fh = G



( f )(h), for all f, h ∈ L 2 (G).

Finally, if f , g ∈ L 2 (G) and γ0 ∈ , testing this identity with f and h = gγ0 which verifies (h)(γ) = (g)(γ0 γ), we get

f g(γ0 ) = ( f ) ∗ (g)(γ0 ), which ends the proof of the Plancherel theorem. 

18

1 A Review of Commutative Harmonic Analysis

1.4.3 A Description of the Wiener Algebra Let us indicate two key consequences (the second of which will be essential in the proof of the Pontryagin duality theorem), which jointly give us a fairly precise description of the algebra A(), which is so far only known to be dense in C0 (), and tell us that the functions of this algebra can be localized: Proposition 1.4.4 The algebra A() consists precisely of convolutions of two functions of L 2 (), namely: H ∈ A() ⇐⇒ H = F1 ∗ F2 , with F1 , F2 ∈ L 2 (). Proof Suppose H is of the previous form. By Theorem 1.4.3, there exist f 1 , f 2 ∈ L 2 (G) such that F1 = ( f 1 ) and F2 = ( f 2 ). Let h = f 1 f 2 ∈ L 1 (G). Theorem 1.4.3 once more gives:  h = ( f 1 ) ∗ ( f 2 ) = F1 ∗ F2 = H, showing that H ∈ A(). The converse is similar, writing h ∈ L 1 (G) as h = f 1 f 2 with f 1 , f 2 ∈ L 2 (G).  Proposition 1.4.5 Let  be a non-empty, open set of . Then, there exists a non-zero function w =  h ∈ A () with support in . Proof Let a ∈  and V a compact neighbourhood of 0 in  such that a + V + V ⊂ . Let χ, χ the indicator functions of a + V and V respectively, so that w = χ ∗ χ is supported  by . By Proposition 1.4.4, w ∈ A(). Moreover, we have by Fubini:  w =  χ  χ = m(V )2 > 0. Therefore w is not zero, which ends the proof. 

1.4.4 A Basic Hilbert-Type Inequality An interesting application of Plancherel’s theorem, which will be of great importance in the study of the space H2 in Chap. 6, and in Chap. 7 as well, is the following:

it Theorem 1.4.6 (Embedding theorem) Let S(t) = ∞ n=1 an n be an absolutely convergent Dirichlet series. Then, we have for any T > 0: 

T −T

|S(t)|2 dt ≤ C

∞ 

|an |2 (T + n)

n=1

where C is a numerical constant. In particular:

(1.4.11)

1.4 The Inversion and Plancherel Theorems



τ +1

|S(t)|2 dt ≤ C

sup τ ∈R

19

τ

∞ 

n|an |2 .

(1.4.12)

n=1

Proof In some sense, the most natural way to proceed would be to use the generalized Hilbert inequality, to which we return just after. But the proof of this inequality, although quite instructive, requires a rather long computation. We therefore give a simple proof of harmonic analysis, due to Gallagher [5]. First, we can assume that T ≥ 1 without loss of generality. Indeed, if this case is settled and if T < 1, we will have 

T

 |S(t)| dt ≤

1

2

−T

≤ 2C

|S(t)| dt ≤ C 2

−1

∞ 

∞ 

|an |2 (1 + n)

n=1

|an |2 × n ≤2C

n=1

∞ 

|an |2 (T + n).

n=1

Let then δ = 1/T ≤ 1, let F be the indicator function of the interval (−2, 0),  (t) = 2eit sin t/t, and let Fδ (t) = F(t/δ)/δ, whose whose Fourier transform is F  (δt) is (uniformly) bounded away from zero on (−T, T ). The Fourier transform F symbol A  B meaning A ≤ cB for some numerical constant c > 0, we clearly have  T  T δ (t)|2 dt |S(t)|2 dt  |S(t)|2 | F −T −T  ∞  ∞ 2  2 δ (t)|2 dt |S(t)| | Fδ (t)| dt = | μ(t)|2 | F ≤ −∞

−∞

where μ is the discrete measure μ =  μ, we get as well  F δ ∗ μ = Fδ ×  

T −T

∞ n=1

 |S(t)|2 dt 

an δlog n . Using Plancherel’s theorem and

∞ −∞

|(Fδ ∗ μ)(x)|2 d x.

Next we have, setting E x = {n; log n − 2δ ≤ x ≤ log n}: (Fδ ∗ μ)(x) =

∞ 

an Fδ (x − log n)

n=1

= δ −1

∞  n=1

 an F

x − log n δ



= δ −1



an

n∈E x

and Cauchy-Schwarz’s inequality gives, |E x | denoting the cardinality of E x ,

20

1 A Review of Commutative Harmonic Analysis



|(Fδ ∗ μ)(x)|2  δ −2

|an |2 × |E x |.

n∈E x

Plugging that inequality into the previous integral, we obtain 

∞ −∞

|(Fδ ∗ μ)(x)| d x  δ 2

−2

∞ 

 |an |

log n

2

|E x |d x.

log n−2δ

n=1

We finally observe that, for a given integer n: x ∈ [log n − 2δ, log n] =⇒ |E x |  nδ + 1.

(1.4.13)

Indeed, m ∈ E x ⇐⇒ e x ≤ m ≤ e2δ e x and since x ≤ log n, there are at most e (e2δ − 1) + 1  nδ + 1 such integers. The value of the integral right of |an |2 is thus  nδ 2 + δ. Multiplying by δ −2 , we get (1.4.11) since δ −1 = T . To get (7.1.6), simply apply the preceding with T = 1/2 1  and bn = an n i(τ + 2 ) . x

Remark The generalized Hilbert inequality mentioned before is due to Montgomery and Vaughan [6] and goes as follows: Theorem 1.4.7 (Montgomery–Vaughan) Let (λn ) be a discrete sequence of distinct real numbers, and δn = inf m=n |λm − λn | > 0, n = 1, 2, . . .. Then, for any finite sequence (an )1≤n≤N of complex numbers, we have:   

 |an |2 am an  , ≤C λ − λn δn n=1 1≤m,n≤N , m 

N

(1.4.14)

m=n

where C is a numerical constant, which can be taken as 23 . As a consequence, if

iλn t is an absolutely convergent Dirichlet series, one gets for T > 0: S(t) = ∞ n=1 an e 

T −T

|S(t)|2 dt 

∞ 

|an |2 (T + δn−1 ).

(1.4.15)

n=1

Inequality (1.4.14) will sometimes be used throughout this book (in particular in Chaps. 6 and 7, as well as in the exercises). Clearly, if we use it with λn = log n and then δn ≈ 1/n, we will obtain a new proof of (1.4.11). Now, if we compute the integral of |S(t)|2 by brute force, we get 

T

−T

|S(t)|2 dt = 2T

∞  n=1

|an |2 +

 am an (ei(λm −λn )T − e−i(λm −λn )T ) . i(λm − λn ) m=n

1.4 The Inversion and Plancherel Theorems

21

Using (1.4.14) with an e±i T λn instead of an gives (1.4.15). This latter inequality can also be obtained by Gallagher’s method. It suffices to replace (1.4.13) by x ∈ [λn − 2δ, λn ] =⇒ |E x | 

δ + 1. δn

(1.4.16)

Let us give the proof of (1.4.16) in the special case when (λn ) is increasing and moreover concave, i.e. λn+1 + λn−1 ≤ 2λn , as this is the case for λn = log n. Then δn = min(λn+1 − λn , λn − λn−1 ) = λn+1 − λn and we have E x = {n − p, n − p + 1, . . . , n} with λ j − λ j−1 ≥ δn , n − p + 1 ≤ j ≤ n so that 2δ ≥ λn − λn− p =

n 

(λ j − λ j−1 ) ≥ pδn ,

j=n− p+1

giving |E x | = p + 1 ≤

2δ δn

+ 1.

1.5 Pontryagin’s Duality Theorem and Applications 1.5.1 The Pontryagin Theorem The successive duals X ∗ , X ∗∗ , . . . of a Banach space X generally form an infinite set (think of the Banach space X = c0 of sequences vanishing at infinity for which X ∗ = 1 , X ∗∗ = ∞ , . . . with an obvious identification), in particular the dual of X ∗ is generally bigger than X , which forces to a sometimes painful distinction between the weak and weak-star topologies. Concerning LCA groups, till now we made a careful distinction between G and its dual  and did not dare to call Fμ the Fourier transform of the measure μ, which it will turn out to be! But fortunately, the sequence of successive duals of a locally compact abelian group (LCA in short) stops in two steps, namely we have the famous Pontryagin theorem, which might be called a Theorem of automatic reflexivity: Theorem 1.5.1 (Pontryagin’s Theorem) Let G be an LCA group and  its dual. Then, = the dual   of  can be identified with G, in other terms the mapping α : G → G  defined by α(x)(γ) = γ(x) ∀x ∈ G, ∀γ ∈  (1.5.1) is an onto isomorphism. Proof It will be relatively short, but many of the preceding results will be needed, and this theorem should be considered as spectacular, useful and deep. We first note that

22

1 A Review of Commutative Harmonic Analysis

α is an algebraic into isomorphism. Indeed, if α(x) = 1, we have γ(x) = 1 ∀γ ∈ , and so x = 1 by the Peter–Weyl theorem. The rest of the proof splits into three steps: 1. α : G →   is an into homeomorphism. Let C be a compact set of  and r > 0. Set M(C, r ) = {x ∈ G; |1 − γ(x)| < r for all γ ∈ C} P(C, r ) = { γ ∈  ; |1 −  γ (γ)| < r for all γ ∈ C}. We have by definition that α [M(C, r )] = P(C, r ).

(1.5.2)

But by Theorem 1.2.2 and Corollary 1.4.2, the sets M(C, r ) and P(C, r ) are a base of neighbourhoods of 0 in G and   , respectively, which proves the claim. In particular, α(G) is topologically and algebraically isomorphic to G and is therefore an LCA group. 2. An LCA subgroup H of an LCA group K is always closed. Indeed, let V a neighbourhood of 0 in  such that V ∩ H is compact and therefore closed in K . Let now a ∈ H and (xα ) a generalized sequence (a filter F) of elements of H converging to a. Let F ∈ F such that α, β ∈ F =⇒ xβ − xα ∈ V ∩ H. Passing to the limit as β → ∞ with α ∈ F fixed, we get a − xα ∈ V ∩ H since V ∩ H is closed. In particular, a − xα ∈ H and, since H is a group, a = a − xα + xα ∈ H . Note that the result is wrong for topological subspaces: the interval ]0, 1[ is locally compact in R, but not closed. 3. α(G) is dense in  . Otherwise, let  be a non-void open set of   such that α(G) ∩  = ∅. By Proposition 1.4.5, there is a non-zero function F ∈ A(  ) supported by  and therefore vanishing on α(G). This function can be written as  F( γ) =



(γ) γ (γ)dγ for some  ∈ L 1 ().

 In particular, 0 = F[α(−x)] =  (γ)γ(x)dγ for all x ∈ G. Setting dμ(γ) = (γ)dγ, this reads Fμ (x) = 0, with the notations of Theorem 1.4.1. We know that this implies μ = 0, i.e.  = 0 and F = 0, which is a contradiction. This ends the proof of Pontryagin’s theorem. 

1.5 Pontryagin’s Duality Theorem and Applications

23

1.5.2 Topological Applications of Pontryagin’s Theorem In this short section, we give two very important consequences of Pontryagin’s theorem. One of them can be used to give an abstract proof of the Kronecker simultaneous approximation theorem, which will turn out to be basic in the later study of Dirichlet series. 1.5.2.1 The first consequence might be called the Hahn–Banach theorem for groups. Let once and for all G be a LCA group,  its dual, and H a closed subgroup of G. The orthogonal H ⊥ of H is the closed subgroup of  defined by H ⊥ = {γ ∈ ; γ(x) = 1 for all x ∈ H }. We first need: Proposition 1.5.2 We always have H ⊥⊥ = H . Proof Clearly, H ⊂ H ⊥⊥ . Let σ : G → G/H the canonical surjection. If x ∈ / H, then σ(x) = 1, and by the Peter–Weyl theorem applied to the LCA group G/H , there exists a character δ on G/H such that δ[σ(x)] = 1. Let γ = δ ◦ σ. Then, γ ∈ H ⊥ and γ(x) = 1, so that x ∈ / H ⊥⊥ . Note that, if H is not closed, we still have H = H ⊥⊥ so that H = G ⇐⇒ H ⊥ = {1}.

(1.5.3) 

Now, we can state Theorem 1.5.3 The following properties hold: (1) (2) (3)

 = H ⊥. G/H  = /H ⊥ . H Each continuous character on H extends to a continuous character on G.

Proof (1) This is a straightforward verification. (2) Replacing G by , H by H ⊥ in (1) and applying Proposition 1.5.2, we get: ⊥ . Now, take the dual of both members and apply the Pontryagin H = /H theorem to the quotient group /H ⊥ . We get exactly the desired conclusion. (3) This is the content of the preceding. Each continuous character on H is given by the class of a continuous character on G, determined modulo H ⊥ . Observe that the conclusion holds even if H is not closed. Indeed, if a character is continuous on H , it is uniformly continuous and can therefore be (uniformly continuously) extended to a character on H , and we are back to the previous situation. 

24

1 A Review of Commutative Harmonic Analysis

1.5.2.1

Bohr Compactification

The second consequence is as follows: forget the topology of , i.e. consider  as a discrete group d equipped with the discrete topology. Then, we know that the

d of d is a compact group, the group of all weak characters δ :  → T. This dual  compact group is denoted by G and is called the Bohr compactification of G. The notation is justified by the Theorem 1.5.4 G is always dense in its Bohr compactification. In other terms, the mapping β : G → G analogous to (1.5.1) defined by β(x)(γ) = γ(x) ∀γ ∈  is a continuous isomorphism and has always dense range. Proof As in the proof of Pontryagin’s theorem, β is an algebraic isomorphism (but in general β(G) is not a LCA group). Let W be a neighbourhood of 0 in G. By Theorem 1.4.4, there exist a compact, i.e. a finite subset C = {γ1 , . . . , γn } of d , and a positive number r > 0, such that V = {x ∈ G; |1 − γi (x)| < r for all i = 1, . . . , n} ⊂ W. Now, the set U = {x ∈ G; |1 − γi (x)| < r for all i = 1, . . . , n} is a neighbourhood of 0 in G such that β(U ) ⊂ V ⊂ W . This shows that β is continuous at the origin, and so at every point. Finally, if x ∈ β(G)⊥ , we have by definition x(x) = 1 for all x ∈ G, i.e. x = 1, showing that β(G)⊥ = {1} and that β(G) = G thanks to (1.5.3).  In the case of the group R which is identified with its dual, the previous general procedure produces the so-called Bohr compactification of R. The functions f : R → C which can be extended to continuous functions F : R → C are called almost periodic functions. This name is justified by the fact, due to Bohr, that they can be characterized as follows: let X = BC(R) be the set of all F : R → C bounded and continuous with the sup-norm F∞ = supt∈R |F(t)| and let ε > 0. A real number τ will be called an ε-period of F ∈ X if we have  Tτ F − F ∞ ≤ ε where Tτ F(x) = F(t + τ ). A subset E ⊂ R will be called relatively dense if there exists a number L > 0 such that each closed interval of length L contains an element of E. Then, a function F ∈ X is called almost periodic if, for any ε > 0, the set E ε of ε-periods of F is relatively dense in R, meaning that there exists some L > 0 such that any interval of length L intersects E ε . As clearly a function is periodic iff the set of its periods is relatively

1.5 Pontryagin’s Duality Theorem and Applications

25

dense, this notion extends that of periodic function and is much more flexible: √for example, the sum of two periodic functions need not be periodic ( f (t) = eit + eit 2 ) whereas the sum (and product) of two almost periodic functions is almost periodic. Moreover, the property is stable under uniform convergence on R, which was one of the motivations of Bohr to introduce the abscissa of uniform convergence σu to be defined later. Finally, it can be shown that Theorem 1.5.5 Let f ∈ X = BC(R). Then, the following are equivalent: (1) f is almost periodic; (2) f is the uniform limit on R of trigonometric polynomials PN , each of the form

PN (t) = Nj=1 a j eiλ j t , λ j ∈ R; (3) f has a continuous extension F to R. Proof • We will admit that (1) =⇒ (2). See ([7],

p. 200) for the proof. • Suppose that (2) holds, and let Q N = Nj=1 a j β(λ j ) ∈ C(R). We have Q N = PN on R. Moreover, since β : R → R has dense range, we also have Q M − Q N C(R) = PM − PN ∞ . By the uniform Cauchy criterion, Q N converges uniformly on R to a continuous function F which extends f . • If (3) holds, we know that F can be approximated uniformly on R by trigonometric polynomials Q N , which can be taken under the form Q N = Nj=1 a j β(λ j ) since

β(R) is dense in R. By restriction, if PN (t) = Nj=1 a j eiλ j t , PN approaches f uniformly on R. Now, if we denote by A P(R) the set of almost periodic functions on R, then A P(R) clearly contains the polynomials PN (for example, by the Dirichlet pigeonhole principle), and is uniformly closed in X , so that f ∈ A P(R). This ends the proof.  The forthcoming theorem will be useful in the study of Hardy spaces of Dirichlet series. We stick to the previous notations. Theorem 1.5.6 Let τ be the Haar measure on R, let f ∈ Y := A P(R) and F its continuous extension to R. Then: T   1 (1) lim T →∞ 2T Fdτ . −T f (t)dt = R F dτ =: (2) For any integer j ≥ 1, we have 1 T →∞ 2T



T

lim

−T

 | f (t)|2 j dt =: f j 22 =

|F|2 j dτ .

iλn t (3) If f (t) = ∞ (with distinct λn ’s) converges uniformly on R, the an ’s n=1 an e  computed by an = R Feλn dτ can also be computed by the formula 1 T →∞ 2T

an = lim



T

−T

f (t)e−iλn t dt, n = 1, 2, . . .

26

1 A Review of Commutative Harmonic Analysis

and we also have 1 lim T →∞ 2T



T

 | f (t)| dt = 2

−T

|F|2 dτ =

∞ 

|an |2 .

n=1

1

(4) lim j→∞ ( f j 2 ) j =  f ∞ . T 1 Proof (1) Let eλ (t) = eiλt , t∈ R. We have lim T →∞ 2T −T eλ (t)dt = 0 if λ  = 0 and = 1 if λ = 0. Similarly, R β(λ)dτ = 0 if λ = 0 and = 1 if λ = 0. Therefore, the relation holds for eλ . By linearity and uniform approximation, it holds for any f ∈ Y. j j (2) Apply (1) to g = f j f ∈ Y and its extension G = F j F . (3) Apply (1) to g = f e−λn , and Parseval’s relation to F=

∞ 

an β(λn ) ∈ L 2 (τ ).

n=1

(4) Let λ =  f ∞ = F∞ . Recall that lim  F  p = F ∞ .

p→∞

(1.5.4)

Indeed, we obviously have lim sup p→∞ F p ≤ λ. Let l < λ. There is a non-void open set U ⊂ R such that |F(x)| ≥ l on U and this set verifies τ (U ) > 0 since the Haar measure is diffuse (an essential fact here). Therefore  1/ p F p ≥ |F(x)| p dτ (x) ≥ l[τ (U )]1/ p U

so that lim inf p→∞ F p ≥ l since τ (U ) > 0. Letting l tend to λ, we get (1.5.4). Now, using (2), we get lim ( f j 2 )1/j = lim F2 j = F∞ =  f ∞ ,

j→∞

ending the proof.

j→∞



Observe that R is not metrizable in view of (1.2.2) since its dual R is not countable. Its Haar measure τ acts on continuous functions F : R → C in a bilateral or unilateral way:   T  1 1 T Fdτ = lim F(t)dt = lim F(t)dt. (1.5.5) T →∞ 2T −T T →∞ T 0 R

1.6 The Uncertainty Principle

27

1.6 The Uncertainty Principle 1.6.1 The Uncertainty Principle on the Real Line Let us here first consider the group R, and for convenience let us renormalize the definition of the Fourier transform, taking  f (ξ) =

 R

e−2iπξt f (t)dt.

This definition has the advantage that f →  f is a unitary operator on L 2 (R), with 2 eigenvalues ±1, ±i. For example,  f = f for the Gaussian function f (x) = e−πx . Now, the so-called uncertainty principle, which became again very popular among mathematicians since the notion of compressed sensing emerged, claims that f and  f cannot be too small together (for example, be both compactly supported), unless f is the zero function. One might say that this uncertainty principle is inscribed in the very definition of the Fourier transform, which involves the product tξ of the “time” variable t and of the “frequency” variable ξ. If we dilate t by a factor λ > 0, we have to compress ξ by the inverse factor 1/λ if we want to keep the product tξ constant: tξ = (λt)(ξ/λ). Equivalently, denoting by gλ the dilate of an arbitrary function g defined by gλ (t) = g(λt), we easily see that 1  ( fλ) = (  f ) λ1 for every λ > 0. λ

(1.6.1)

In particular, if a > 0, we have ξ2 1 −πax 2 = √ e−π a . e a

This is a limiting case as shown by a beautiful theorem of Hardy [8]: Theorem 1.6.1 (Hardy) Let f ∈ L 1 (R). Suppose that f and  f are both small at infinity in the following sense: f (x) = O(e−πax ), 2

2  f (ξ) = O(e−πbξ ), with a, b > 0.

Then, ab > 1 =⇒ f = 0. More precisely, ab = 1 =⇒ f (x) = Ce−πax . 2

We will not treat here the equality case. The initial proof of Hardy made use of refined versions of the maximum modulus principle. We will instead give a recent and quite elementary proof due to Hedenmalm [9] of a form of the uncertainty principle due to Beurling and Hörmander (see [10]), independently.

28

1 A Review of Commutative Harmonic Analysis

Theorem 1.6.2 (Beurling–Hörmander) Let f ∈ L 1 (R). Suppose that   R

R

| f (x)  f (y)|e2π|x y| d xd y < ∞.

(1.6.2)

Then, f = 0. Proof Let us begin with observing that this theorem is stronger than Hardy’s uniqueness one. For if the assumptions of the latter are verified, the discriminant 1 − ab of the quadratic form ax 2 − 2x y + by 2 is negative, and ax 2 − 2x y + by 2 ≥ c(x 2 + y 2 ) for some positive constant c > 0, so that for some other constant C > 0:   R

R

| f (x)  f (y)|e2π|x y| d xd y ≤ C

 

e−π(ax

2

−2|x y|+by 2 )

e−c(x

2

+y 2 )

R R  

≤C

R

R

d xd y

d xd y < ∞

and f = 0 in view of (1.6.2). Let us now come to the proof of Theorem 1.6.2. We first observe that, since e2π|x y| ≥ 1, we have as well      | f (x)|d x × | f (y)|dy = | f (x)  f (y)|d xd y < ∞, R

R

R

R

 f ∈ L 1 (R), so that f (x) = R  f (y)e2iπx y dy for almost all x, and i.e. f ∈ L 1 (R) and  we may assume from the beginning that f is continuous, bounded and integrable as well as  f . We then introduce the function   F(λ) =

R

R

f (x)  f (y)e2iπλx y d xd y.

(1.6.3)

The assumptions obviously imply that F is holomorphic in the strip S = {λ; |Im λ| < 1} and continuous and bounded in its closure. Moreover, the Fourier inversion formula gives   f (y)e2iπλx y dy = f (λx), λ, x ∈ R, R

so that, for real arguments, F may be expressed in the form  F(λ) =

R

f (x) f (λx)d x, λ ∈ R.

Equation (1.6.4) clearly implies, by a change of variable,

(1.6.4)

1.6 The Uncertainty Principle

29

F(λ) =

 1  F 1/λ , λ ∈ R∗ = R\{0}. |λ|

(1.6.5)

To get rid of the highly √ non-analytic modulus in the previous functional equation, we consider J (λ) = 1 + λ2 (which defines an analytic function in C deprived of the two half-lines [i, i∞[ and ] − i∞, −i], with value 1 at λ = 0). And we set  = F J . This function is well-defined and continuous on R, and holomorphic in a neighbourhood of D\{±i}, where D denotes the unit disk. The functional equation (1.6.5) becomes, as is readily checked   (λ) =  1/λ , λ ∈ R∗ .

(1.6.6)

Let now T\{±i} ⊂ U ⊂ S\{0} be an open set with two connected components. We can extend (1.6.6) as follows:   (λ) =  1/λ , λ ∈ U.

(1.6.7)

Indeed, the two members of (1.6.7) are holomorphic in U and, by (1.6.6), coincide on (±1 − δ, ±1 + δ) ⊂ U for some positive δ. The end of the proof now relies on the following lemma: Lemma 1.6.3 We have (λ) = 0 for all] λ ∈ D. Once we have Lemma 1.6.3, the proof is finished. Indeed, we see that F(λ) = 0 for all λ ∈ D. In particular, F(1) = R | f (x)|2 d x = 0, implying f = 0. It remains to prove the lemma, what we do now. The construction will be illustrated by a picture (Fig. 1.1).

Fig. 1.1 The set U

30

1 A Review of Commutative Harmonic Analysis

Set  = D\ {±i} and consider the function  defined by ⎧ ⎪ ⎨ (λ)   if λ ∈  (λ) :=  1/λ if |λ| > 1 ⎪ ⎩ 0 if λ = ±i It follows from (1.6.7) (this is the Schwarz reflection principle through the unit circle) that  is holomorphic in C\{±i} and continuous on C. Indeed, this is implied by (1.6.7) and the fact that, F being bounded in S, we have lim (λ) = 0 = (±i).

λ→±i, λ∈D

By Riemann’s theorem on removable singularities, we get that  is an entire function. Moreover, this function is bounded in C by its very definition. Liouville’s theorem now gives (λ) = c0 where c0 is a constant, and moreover c0 = (i) = 0. This ends the proof of the Beurling–Hörmander Theorem. 

1.6.2 The Uncertainty Principle on Finite Groups In case of finite groups, the (discrete) uncertainty principle says that f and  f cannot have supports of small cardinality together unless f = 0. We focus now on a form of this principle mimicking the beautiful presentation by Tao [11]. Let G be a finite group and recall that its dual group Gˆ is isomorphic to G. If f = 0 is a complex  function defined on G, fˆ denotes the discrete Fourier transform of f defined on G by 1   f (x)χ(x). f (χ) = |G| x∈G We denote by |A| the cardinality of the finite set A, and by supp f the set of x ∈ G such that f (x) = 0. The first version of the uncertainty principle, proved in [12], is a direct consequence of the Cauchy–Schwarz inequality and Parseval’s identity which follows:  1  | f (χ)|2 = | f (x)|2 . (1.6.8) |G|  x∈G x∈G

Theorem 1.6.4 Let f = 0 be a complex function defined on G. Then |supp f | × |supp  f | ≥ |G|. Proof Observe that

(1.6.9)

1.6 The Uncertainty Principle

31

  1  |supp f | 1/2   2 1/2  f |1 := | f (x)| ≤ | f (χ)| . |G| x∈G |G|  χ∈G

Indeed, we apply Cauchy–Schwarz’s inequality to the inner product: 1/2   1 1/2  1   | f (x)| 1 2 ≤ | f (x)| |G|1/2 |G|1/2 |G| x∈G |G| x∈supp f x∈supp f and by using (1.6.8), we get  f 1 =

1  | f (x)| ≤ |G| x∈G



But now,

 χ∈G

| f (χ)|2 =



|supp f | |G|



1/2  

| f (χ)|2

1/2

.

χ∈supp  f

| f (χ)|2 ≤ |supp  f|  f 2∞

χ∈supp  f

and at this point,   f 1 ≤

1/2 1/2  |supp f | |supp  f| |supp f | |supp  f|  f ∞ ≤  f 1 |G| |G|

which ends the proof since  f 1 = 0.



The previous theorem admits the following improvement due to Tao: Theorem 1.6.5 Let p be some prime number and f : G := Z/ pZ → C, f = 0. Then |supp f | + |supp  f | ≥ p + 1. (1.6.10) Observe that this inequality provides an improvement of (1.6.9) when G = Z/ pZ since the hyperbola x y = p remains under its secant x + y = p + 1 between the points (1, p) and ( p, 1).  we simply have for Proof First of all, note that in the case G = Z/ pZ = G, k ∈ Z/ pZ 1   f ( j)ω − jk f (k) = p 0≤ j≤ p−1 with ω = e2iπ/ p . We shall prove that if (1.6.10) does not hold for some f , then f must be zero. Let us put A = supp f , B = supp  f and assume that |A| + |B| ≤ p. If |A| = p − t with t ≥ 0, B has at most t elements so that p − t elements can be chosen in  G\B; we have obtained in this way a subset A of Z/ pZ such that:

32

1 A Review of Commutative Harmonic Analysis

(i) |A | = |A|, (ii)  f (k) = 0 for every k ∈ A . Consider the square matrix T := T A,A whose entries are T jk =

1 − jk ω , |A|

j ∈ A, k ∈ A .

/ A by definition and if k ∈ A , Clearly, T f |A = 0: indeed, T f | A (k) = 0 if k ∈ T f |A (k) =

p 1 p 1  p  f |A ( j)ω − jk = f ( j)ω − jk = f (k) = 0, |A| p j∈A |A| p j∈Z/ pZ |A|

by (ii). But T is invertible since this is exactly what Chebotarev’s lemma says Let p be some prime number and ω = e2iπ/ p . For any subsets A, A of Z/ pZ with |A| = |A |, the matrix (ω jk ) j∈A,k∈A is invertible.  Finally, f |A = 0 and f = 0. Before proving Chebotarev’s lemma we add a remark: if |A| + |B| ≥ p + 1, there exists f : Z/ pZ → C, f = 0, such that supp f = A and supp  f = B, in other words (1.6.10) is optimal. Assume without loss of generality that |A| + |B| = p + 1; then there exists A ⊂ Z/ pZ such that |A | = |A| and A ∩ B = {k  }, that is A = B c ∪ {k  }. Let us choose g such that g(k) = 0 for k ∈ B c but g(k  ) = 0. Observe that suppg ⊂ B. Since T := T A,A is invertible as we already noticed, T −1 g|A =: f is supported by A and g =  f . It remains to prove that supp f = A, supp  f = B. But p + 1 ≤ |supp f | + |supp  f | = |A| − |A\supp f | + |B| − |B\supp  f| ≤ p + 1 − |A\supp f | − |B\supp  f| so that

|A\supp f | = |B\supp  f| = 0

whence the remark. We turn back to the fundamental Chebotarev lemma. Lemma 1.6.6 (Chebotarev) If p is some prime number and ω = e2iπ/ p , then any minor of the Schur matrix (ω jk )0≤ j,k≤ p−1 is invertible. Proof Let us fix n < p and consider any n × n-minor of the Schur matrix. It amounts to choosing n columns from the following rectangle:

1.6 The Uncertainty Principle

33

⎛

ω 2k1 ω 2k2 ··· ω 2kn

1 ω k1 ⎜ 1 ω k2 ⎜ ⎝ · ··· 1 ω kn ⎛

that we rather write

1 z1 ⎜ 1 z2 ⎜ ⎝ · ··· 1 zn

⎞ ··· ···⎟ ⎟ ···⎠ ···

⎞ z 12 · · · z 22 · · · ⎟ ⎟ ··· ···⎠ z n2 · · ·

with evident notations. Thus the selected minor may have the form: ⎛

j

z 11 ⎜ .. ⎝ . j z n1

··· .. . ···

j ⎞ z 1n .. ⎟ . . ⎠

(1.6.11)

j z nn

We denote by D(z 1 , . . . , z n ) its determinant and we first observe that 

D(z 1 , . . . , z n ) = P(z 1 , . . . , z n )

(z j  − z j ),

1≤ j< j  ≤n

where P ∈ Z[X 1 , . . . , X n ]. In order to get the invertibility of the minor, we have just to prove that P is not zero. We shall make use of the following: Lemma 1.6.7 We fix P ∈ Z[X 1 , . . . , X n ] and assume that P(ω1 , . . . , ωn ) = 0, where ω1 , . . . , ωn are any given pth roots of the unit. Then P(1, . . . , 1) ≡ 0 mod p. Admitting this result, we are led to catch P(1, . . . , 1), what we dothrough successive differentiations of D, finally calculated at (1, . . . , 1). The term 1≤ j< j  ≤n (z j  − z j ) is vanishing at (1, . . . , 1) as well as its successive derivatives till the constant term; besides,  d  (z ) j (z − 1) j = j! z=1 dz We thus consider the operator D = (z 1

∂ 0 ∂ 1 ∂ n−1 ) (z 2 ) · · · (z n ) =: 0 1 · · · n−1 ∂z 1 ∂z 2 ∂z n

and we compute (D D)(1, . . . , 1). It is easy to get (D D)(1, . . . , 1) = P(1, . . . , 1)(n − 1)!(n − 2)! · · · 1! Now, since n < p, P(1, . . . , 1) ≡ 0 mod p if (and only if) (D D)(1, . . . , 1) is not a multiple of p.

34

1 A Review of Commutative Harmonic Analysis

In order to check this last point, we make use of the precised form (1.6.11) of the minor. The same computation now gives, via  j (z k ) = k j z k : (D D)(1, . . . , 1) = det( jm−1 )1≤,m≤n and this van der Monde determinant can never be a multiple of p if n < p. We are left with the proof of the auxiliary lemma. Let us assume that P(ω k1 , . . . , ω kn ) = 0 with ω = e2iπ/ p , and let us put R(z) = P(z k1 , . . . , z kn ); thus R vanishes at ω and we aim to conclude R(1) ≡ 0 modulo p. Without loss of generality, we may assume that the degree of R is bounded by p − 1: if not, we consider the euclidean division of R by z p − 1; the remaining polynomial, say T , has still integral coefficients and satisfies T (ω) = 0 as well. But now this implies that R(z) = λ(1 + z + · · · + z p−1 ), λ ∈ Z, and R(1) = λ p which proves the auxiliary lemma and the Chebotarev lemma in the wake.



1.7 Exercises 1. (Steinhaus theorem) (i) Let G be a sigma-compact LCA group, A, B ⊂ G Borel sets of positive measure. Show that the Minkowski sum A + B = {a + b; a ∈ A and b ∈ B} has interior points (Hint: if m(A) and m(B) < ∞, consider the convolution 1 A ∗ 1 B and its Fourier transform). (ii) Let (an ) be a sequence of real numbers such that eian t → 1 point-wise on R. ∞ Show that an → 0 (Hint: consider 0 e−(1+ian )t dt). (iii) Let (an ) be a sequence of real numbers such that eian t converges for t ∈ A ⊂ R, where m(A) > 0. Show that the convergence holds for all t ∈ R, that (an ) is a Cauchy sequence, and therefore converges. Extend the result to other LCA groups. 2. Let E be a finite-dimensional real normed space, and G the (compact) group of its linear isometries, which admits a left and right translation-invariant Haar measure m (even if it is not commutative). Fix a ∈ E with a = 1. (i) Show that there exists on E a positive-definite quadratic form Q such that Q(a) = 1 and which is G-invariant: Q(gx) = Q(x) for all x ∈ E, g ∈ G. (Hint: start from some positive-definite quadratic form Q 0 on E and average it with respect to m). (ii) Assume that G acts transitively on the unit sphere of E. Show that

1.7 Exercises

35

x2 = Q(x) for all x ∈ E. In other terms, E is a Hilbert space. The extension of this result to the case of an infinite-dimensional space is open.  3. Give an example of an LCA group G and of a function f ∈ L 1 (G) such that  f ∗  f˜1 <  f 21 (Hint: consider f (z) = 1 + r z, r > 0, on T or f = 1(0,1/2) − 1(−1/2,0) , on R). 4. Let G be a compact, connected and metrizable abelian group with Haar measure m, and  its dual, which is a countable discrete group. (1) Show that if γ ∈  and γ = 1, the kernel Hγ = {x ∈ G; γ(x) = 1} of γ has an empty interior. (2) Using the Baire category theorem, show that there exists a ∈ G such that: γ ∈  and γ = 1 =⇒ γ(a) = 1. (3) Using the Peter–Weyl theorem, show that if f : G → C is continuous, we have  N 1  f (a n ) → f dm N n=1 G and by approximation, formulate an analogous result if f is the indicator function of an open set. (4) Conclude in several ways that the subgroup generated by a is dense in G, which is thus seen to be topologically cyclic (or monothetic). 5. Let P the set of almost periodic polynomials P(t) =

N 

an eiλn t , with an ∈ C, λn ∈ R, t ∈ R.

n=1

We set, for P ∈ P and 1 ≤ p < ∞: P p =

 lim

T →∞

1 T



T

|P(t)| p dt

 1p

.

0

Show that the limit indeed exists, and that it defines a norm on P. N n it , show that the expo6. By considering the Dirichlet polynomial P(t) = n=1 1 nent one of n in the right-hand-side of (1.4.12) is sharp (Hint: 0 |P(t)|2 dt # N 2 / log N ). 7. Let a, b > 0 such that ab < 1. Show that there is an infinite-dimensional vector space V ⊂ L 1 (R) ∪ L 2 (R) such that, for any f ∈ V , one has at infinity: f (x) = O(e−πax ), 2

2  f (ξ) = O(e−πbξ ).

36

1 A Review of Commutative Harmonic Analysis

What can you say in the case ab = 1? 8. (Cauchy–Davenport theorem) Let G be a finite abelian group and for f , g complex functions defined on G, let us consider the convolution product ( f ∗ g)(x) =

1  f (x − y)g(y), x ∈ G. |G| y∈G

Recall that supp( f ∗ g) ⊂ supp f + supp g. We specialize: G = Z/ pZ with p some prime number, and we intend to prove the following, where |E| denotes the cardinality of the finite set E: If A and B are non-empty subsets of Z/ pZ, such that |A| + |B| ≥ p + 1 then A + B = Z/ pZ. (a) For every pair of subsets X, Y of Z/ pZ with |X | + |A| = p + 1 and |Y | + |B| = p + 1, we can choose f and g such that f = X, supp g = Y. supp f = A, supp g = B, supp  (b) Prove that |A + B| + |X ∩ Y | ≥ p + 1 and conclude with a suitable choice of X and Y . (c) More precisely, for A and B non-empty subsets of Z/ pZ, can you establish |A + B| ≥ min{|A| + |B| − 1, p}?

References 1. D. Li, H. Queffélec, Introduction à l’étude des espaces de Banach, Analyse et probabilités, Cours spécialisés SMF, no 12 (2004) 2. D.L. Cohn, Measure Theory, Birkhäuser (Mass, Boston, 1980) 3. B. Host, J.F. Mela, F. Parreau, Analyse harmonique des mesures, Astérisque 135–136, Société mathématique de France (1986) 4. W. Rudin, Fourier Analysis on Groups, 2nd edn. (John Wiley and Sons, 1967) 5. P.X. Gallagher, A large sieve density estimate near σ = 1. Inventiones Math. 11, 329–339 (1970) 6. H. Montgomery, B. Vaughan, Hilbert’s inequality. J. Lond. Math. Soc. 2, 73–82 (1974) 7. Y. Katznelson, An Introduction to Harmonic Analysis, 3rd edn. (Cambridge University Press, 2004) 8. G.H. Hardy, A theorem concerning Fourier transforms. J. Lond. Math. Soc. 8(3), 227–231 (1967) 9. H. Hedenmalm, Heisenberg’s uncertainty principle in the sense of Beurling, J. Anal. Math. 118(2), 691–702 (2012) 10. L. Hörmander, A uniqueness theorem of Beurling for Fourier transforms pairs, Ark. Mat. 29(2), 237–240 (1991) 11. T. Tao, An uncertainty principle for cyclic groups of prime order. Math. Res. Lett. 12, 121–127 (2005) 12. L. Donoho, B. Stark, Uncertainty principles and signal recovery, S.I.A.M. J. Appl. Math. 49, 906–931 (1989)

Chapter 2

Ergodic Theory and Kronecker’s Theorems

2.1 Elements of Ergodic Theory Measure theory, sometimes, brings out the existence of specific elements by giving a positive measure to the set of such objects. For want of anything better, it can also be used in the number-theoretical framework to produce classifications of real numbers through their expansions. Ergodic theory will play a role in this purpose.

2.1.1 Basic Notions in Ergodic Theory We shall need classical concepts of integral, measure and probability, as explained in [1], and we just give a short introduction to dynamical systems. In this account, (X, A, μ) denotes a measure space, a probability space when μ is a probability measure on X . If μ and ν are two positive measures on X , we say that μ is absolutely continuous with respect to ν and we write μ  ν if negligible sets for ν are also negligible for μ, i.e.: A ∈ A and ν(A) = 0 =⇒ μ(A) = 0. Those measures are said to be “equivalent” if both μ  ν and ν  μ hold. On the opposite, we say that μ and ν are orthogonal and we write μ ⊥ ν if one can find a set A ∈ A such that μ(A) = 0 and ν(Ac ) = 0. Dynamics focuses on time-dependent phenomena, whose simplest model consists in the iteration of some transformation on X , which is a measurable mapping T : X → X. Definition 2.1.1 We say that the transformation T preserves μ on X if μ = T (μ), the image of μ under T , in other words, © Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_2

37

38

2 Ergodic Theory and Kronecker’s Theorems

  μ T −1 A = μ ( A) , ∀A ∈ A. For physical reasons, only measure-preserving transformations have been considered to be relevant and we say as well that μ is T-invariant. Definition 2.1.2 A dynamical system (X, A, μ, T ) is a probability measure space endowed with a measure-preserving transformation T. By the study of the dynamics of T or, more briefly, the study of the dynamical system, we mean the description of the orbits {T n x, n ≥ 0} of elements x ∈ X , and, more generally, the behaviour of the sequence ( f (T n x))n≥0 for any measurable f . The functions X n = f ◦ T n appear as identically distributed but dependent random variables, and the first weak notion of independence is now introduced. Definition 2.1.3 Let (X, A, μ, T ) be a dynamical system. Then T is called ergodic (or μ is T-ergodic) if the invariant sets of A are trivial in the following sense: A ∈ A and T −1 A = A =⇒ μ (A) = 0 or 1. Roughly speaking, (X, A, μ, T ) cannot be decomposed into metrically disjoint dynamical systems and it is referred to as an ergodic dynamical system. Moreover, it is called uniquely ergodic if μ is the unique probability measure preserved by T and this property will turn out to be powerful.

2.1.2 Ergodic Theorems The very useful ergodic theorem appears as a generalization of the strong law of large numbers, dealing with dependent variables as we already pointed out; historically, the first step is due to von Neumann. Theorem 2.1.4 (von Neumann) Let (X, A, μ) be a probability measure space and T a measure-preserving and (invertible) transformation of (X, μ); then, for any f ∈ L 2 (X ), 1  lim f ◦ T n = P f in L 2 (X ) N →∞ N nN

Let us fix now ε > 0 and F ∈ L 2 (X ) such that h 0 1 :=  f − F1 ≤ ε/2. Then, in the same spirit, we split F = F0 + (g − T g) + h 1 with F0 ∈ HI , g, h 1 ∈ L 2 (X ) and h 1 2 ≤ ε/2. Thus, f = F0 + g − T g + h, h = h 0 + h 1 , h1 ≤ ε and, for n ≥ N ≥ 1, An f − A N f =

  1 1  g − T ng − g − T N g + An h − A N h. n N

40

2 Ergodic Theory and Kronecker’s Theorems

We shall decompose the set {supn>N |An f − A N f | > α} we are interested in; obviously, by taking the complement of the sets below, we have the following inclusion: {sup |An f − A N f | > α} ⊂ ∪n>N { n>N

1 |g − T n g| > α/4} ∪ {sup An |h| > α/4} N n≥1

so that μ({sup |An f − A N f | > α}) ≤ 

n>N

μ({|g − T g| > nα/4}) + μ({sup An |h| > α/4}). n

n≥1

n>N

The first term in the RHS can easily be bounded as follows: 

μ({|g − T n g| > nα/4}) ≤

n>N

2 4  1

C

n

− T g

g

≤ 2 α2 n>N n 2 N

where C = C(α, ε). We are thus left to control the sequence (A N |h|). The following “maximal” lemma will give the conclusion. Lemma 2.1.6 Let ϕ ≥ 0 and ϕ∗ := supn≥1 An ϕ. Then we have for any α > 0 



1 μ ϕ >α ≤ α ∗

 {ϕ∗ >α}

ϕ dμ ≤

1 ϕ1 . α

Admitting this inequality that we apply with ϕ = |h|, we finally get μ({sup |An f − A N f | > α}) ≤ n>N

4 4ε C C + h1 ≤ + N α N α

Set temporarily L = lim sup μ({sup |An f − A N f | > α}). N →∞

n>N

We first get, letting N → ∞ : L ≤ 4ε/α. And finally L = 0 since ε > 0 is arbitrarily small. This was the point to be proved. It remains to prove the maximal Lemma 2.1.6, which appears as a direct consequence of a more general one. We introduce a notation: S0 f = 0, Sk f = f + T f + · · · + T k−1 f, Mn f = max Sk f. 1≤k≤n

Lemma 2.1.7 Let f be a real function in L 1 (X ) and E n := {Mn f > 0}. Then,  f dμ ≥ 0. En

2.1 Elements of Ergodic Theory

41

Proof We start with a consequence on (Mn f ) of the cocycle relation for the Birkhoff sums: since Sk f = f + T Sk−1 f and S0 f = 0, we have Mn f := max( f, f + T f, . . . , f + T Sn−1 f ) f + max(0, T f, . . . , T Sn−1 f ) f + max(0, T Mn−1 f ).

= =

By definition of the sets (E n ), T Mn−1 f is > 0 on T −1 E n−1 precisely, so that 





Mn f dμ = En

f dμ + En

E n ∩T −1 E n−1

T Mn−1 f dμ;

it follows that 





f dμ ≥ En

Mn f dμ − 

En



En

T −1 E n−1

Mn f dμ −

=

T Mn−1 f dμ



Mn−1 f dμ E n−1

(Mn f − Mn−1 f ) dμ

≥ E n−1

since E n−1 ⊂ E n and Mn f > 0 on E n−1 . We deduce the result from the increasing  character of the sequence (Mn f ). Lemma 2.1.6 is now easily derived from the previous one. Given ϕ and α > 0, we simply apply Lemma 2.1.7 with f = ϕ − α; it is clear that E n is nothing but {max1≤k≤n Ak ϕ > α} and the conclusion results in  {max1≤k≤n Ak ϕ>α}

ϕ dμ ≥ αμ({ max Ak ϕ > α}). 1≤k≤n

Letting n → ∞ we get the claimed inequality with help of the Beppo Levi theorem since ϕ ≥ 0 and max1≤k≤n Ak is increasing to ϕ∗ .  The proof of the theorem is complete.  This theorem has many consequences: in number theory when it is used with a specific ergodic dynamical system (X, μ, T ), for example, the Gauss dynamical system, or more conceptually when it is applied to a transformation admitting a lot of invariant and ergodic probability measures. We list below some of them which will be useful later. Corollary 2.1.8 Two distinct probability measures on X which are preserved by the same transformation T and ergodic must be orthogonal. Proof If μ and ν are two distinct T -invariant measures, one can find a set B ∈ A such that μ(B) = ν(B); applying the Birkhoff theorem with f = 1 B on both dynamical

42

2 Ergodic Theory and Kronecker’s Theorems

systems (X, μ, T ) and (X, ν, T ), we get that and to ν(B) ν-a.e. It follows that

1 N

 n 0, we set O(u, λ, ε) = {ϕ ∈ CR (E); sup |eiλϕ(x) − u(x)| < ε}, x∈E  O(u, ε) = O(u, λ, ε). λ>0

Clearly, O(u, λ, ε) and O(u, ε) are open sets. As is usual when applying Baire’s category theorem, the main point is to show For each u ∈ S(E) and ε > 0,

O(u, ε) is dense in CR (E).

(2.4.1)

Once we know (2.4.1), let (u n ) be a countable, dense, subset of S(E), each function appearing infinitely often in  the sequence, and let (εn ) a sequence of positive numbers with limit 0. The set A = n O(u n , εn ) is a dense G δ - set of CR (E) by Baire’s

2.4 Towards Infinite Dimension

63

theorem. One usually says that quasi-all elements of CR (E) are in A. Let ϕ ∈ A. Suppose that ϕ(x) = ϕ(y). For any integer n, there is a positive real number λn such that |eiλn ϕ(z) − u n (z)| < εn for each z ∈ E. So that |u n (x) − u n (y)| ≤ |u n (x) − eiλn ϕ(x) | + |u n (y) − eiλn ϕ(y) | < 2εn . Now, let u ∈ S(E). Taking a sequence n j → ∞ such that u n j → u uniformly, and passing to the limit, we get u(x) = u(y). By Urysohn’s lemma, this implies that x = y, so that ϕ is injective, and that ϕ(E) is homeomorphic to E. Back to (2.4.1): since E is completely disconnected and compact, the connected component of a point a ∈ E is the intersection of open and closed (clopen) subsets containing a (Bourbaki, Topologie Générale, pp. 224–225), in particular a has a basis of clopen neighbourhoods, which implies that, for any r > 0 E admits a disjoint clopen cover (E j ) with diam E j < r for each j,

(2.4.2)

(meaning that the topological dimension of E is 0). Let now δ and ε be positive numbers, ψ ∈ CR (E), u ∈ S(E). We have to find some ϕ ∈ CR (E) such that: (1) ϕ − ψ∞ ≤ δ; this will be the case if ϕ(x) = ψ(x) + c j on E j , where c j is a constant such that sup j |c j | ≤ δ. Note that ϕ is continuous since the E j ’s are clopen. (2) supx∈E |eiλϕ(x) − u(x)| < ε for some λ > 0. Fix λ ≥ 2π/δ. Thanks to (2.4.2), we can choose r and the E j ’s in order that the oscillation of u and eiλψ on each E j is < ε/2. Let u j = eiα j and v j = eiβ j be the values, Respectively, taken by u and eiλψ on E j , with max(|α j |, |β j |) ≤ π. We now adjust c j in order to have u j e−iλc j = v j , for example c j =

αj − βj . λ

Observe that this c j verifies: |c j | ≤ 2π/λ ≤ δ. Moreover, if x ∈ E j , one has ε 2 ε ε = |eiλψ(x) − u j e−iλc j | + ≤ |eiλψ(x) − v j | + |v j − u j e−iλc j | + 2 2 ε iλψ(x) = |e − v j | + < ε. 2

|eiλϕ(x) − u(x)| ≤ |eiλϕ(x) − u j | + |u j − u(x)| < |eiλϕ(x) − u j | +

This shows (2.4.1). Moreover, we see that ϕ ∈ A =⇒ ϕ(E) = F is a Kronecker set. Indeed, let v ∈ S(F). Then, v ◦ ϕ ∈ S(E), therefore, there exists some λ > 0 such that supx∈E |v ◦ ϕ(x) − eiλϕ(x) | < ε, that is

64

2 Ergodic Theory and Kronecker’s Theorems

sup |v(y) − eiλy | < ε. y∈F

This ends the proof of Theorem 2.4.6, even showing that quasi-all continuous images of E are Kronecker sets. 

2.5 Exercises 1. Show that the numbers 1, log p1 , . . . , log pd are independent for each d. 2. Show that an independent compact set in an LCA group has Haar measure zero. 3. Let A, B be two Borel subsets of R+ . Show that d(A ∩ B) ≥ d(A) + d(B) − 1. 4. Let E be a compact, metric space. Show that the set S(E) is separable. 5. Let K ⊂ T be the compact set formed by the sequence (e2iπxn )n≥1 and its limit, where xn = α + β n , with 0 < β√< 1, the real numbers 1, α, β n , n = 1, 2, . . . being independent (for example, α = 2 and β = 1/e). (1) Show that K is an independent subset of T. (2) Show that a Kronecker L ⊂ T set is a Dirichlet set, namely, there exists a sequence of positive integers λ j → +∞ such that z λ j → 1 uniformly for z ∈ L (Hint: if u ∈ S(E) is not a character, there exists a sequence μ j of integers with μ j → +∞ and μ j+1 − μ j → +∞, such that z μ j → u(z) uniformly or z μ j → u(z) uniformly. Consider λ j = μ j+1 − μ j ). (3) Show that the set K above is not a Dirichlet set, and therefore not a Kronecker set (Hint: if λ j ≥ 1/β 3 , there is an integer n ≥ 1 such that 1/β 2 ≤ β n λ j ≤ 1/β ). 6. Prove that an equidistributed subsequence can be extracted from any sequence of real numbers which is dense in [0, 1]. 7. Let (λn )n≥1 be a sequence of real numbers such that: λn+1 − λn ≥ δ for every n ≥ 1, where δ > 0 is independent of n. We shall prove that (λn x)n≥1 is uniformly distributed modulo 1 for almost all real x. We recall the generalized Hilbert inequality (see Chap. 1):    

 1≤m,n≤N m=n

 N z m z n  π  ≤ |z n |2 , λm − λn  δ 1

for every N ≥ 1 and z 1 , . . . , z N ∈ C. (1) Fix two real numbers a < b, an integer N ≥ 1, and put M N (x) = b Show that a |M N (x)|2 d x ≤ (b − a + δ −1 )/N . (2) Deduce the result.

1  N e2iπλ j x . j=1 N

2.5 Exercises

65

8. (Van der Corput sets) In what follows, T = R/Z is identified with [0, 1) and M + (T) denotes the set of positive measures on T. For  ⊂ N, we consider the two following properties: (P1 ) ∀σ ∈ M + (T), σ(h) ˆ = 0 ∀h ∈  =⇒ σ({0}) = 0. (P2 ) For all real sequence (u n ), (u n+h − u n ) uniformly distributed mod.1 ∀h ∈  =⇒ (u n ) uniformly distributed mod.1. A set  enjoying (P2 ) is called a van der Corput set. (1) Prove that (P1 ) =⇒ (P2 ). (2) Let σ ∈ M + (T) and Q = Q ∩ [0, 1). For every ε > 0, one can find m > 0 such { j/m, j = 0, 1, . . . , m − 1}. that σ(Q\Qm ) ≤ ε, where Qm = 2 2 (3) We fix m and we put f N (t) = N1 1≤n≤N e−2iπn m t . Prove that  lim

N →0 Qc

f N (t) dσ(t) = 0.

(4) Suppose now that σ(k ˆ 2 ) = 0 for every k = 0. By a suitable decomposition of  f (t) dσ(t), show that σ({0}) = 0. Which result is thus obtained ? T N 9. Let (X, B, μ, T ) be an invertible dynamical system and consider A ∈ B with positive measure. (1) Why does there exist a positive measure σ ∈ M + (T) such that  σ (k) = μ(A ∩ T

−k

 A) =

1 A · 1 A ◦ T k dμ, ∀k ∈ Z? X

 (2) Show that lim N →∞ N1 n 0. 10. (Ergodicity) Let (X, B, μ, T ) be a dynamical system. Prove the equivalence of the following assertions: (i) The system is ergodic. (ii) For all A, B ∈ B, the sequence μ((T −n A) ∩ B) converges to μ(A)μ(B) in Cesàro mean.     (iii) For all f, g ∈ L 2 , N1 n 1/2, X U must be finite. 12. Prove that the dynamical system (T, T, m), where T is the 2-shift transformation, is ergodic and semi-conjugate to the unilateral shift on the dyadic expansion of real numbers in [0, 1). Prove that it is not uniquely ergodic.  13. Let P be a subset of the set of prime numbers such that p∈P 1/ p diverges. We define  K as the set of squarefree products of elements of P. We fix M and put PM = p≤M p. p∈P

(a) By involving the Möbius function μ, observe that 

1=

n≤N (n,PM )=1

μ(d) =

n≤N d|n, d|PM

Deduce that lim



N →∞

1 N

 n≤N (n,PM )=1

 d|PM

1=



 N . μ(d) d 

(1 −

p≤M p∈P

1 ). p

(b) We consider a sequence of real numbers (u n ) with the property that (u kn ) is uniformly distributed modulo 1 for every k ∈ K . Prove that the sequence itself is uniformly distributed modulo 1. (Hint: for h = 0, split the sum N  n=1

e(hu n ) =

 n≤N (n,PM )=1

e(hu n ) +





e(hu n ).)

k|PM n≤N k≥2 (n,PM )=k

References 1. W. Rudin, Real and Complex Analysis, 3rd edn. (1984) 2. A. del Junco, J. Rosenblatt, Counterexamples in ergodic theory and number theory. Math. Ann. 245, 185–197 (1979) 3. A. Granville, Z. Rudnick, Equidistribution in number theory, an introduction. NATO Sci. Ser. II Math. Phys. Chem. 237 (2007) (Springer, Dordrecht) 4. M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.-P. Schreiber, Pisot and Salem Numbers (Birkhäuser Verlag, Basel, 1992). With a preface by David W. Boyd 5. C. Pisot, La répartition modulo 1 et les nombres algébriques. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 205–248 (1938) 6. R. Salem, Algebraic Numbers and Fourier Analysis (D. C. Heath and Co., Boston, Mass, 1963)

References

67

7. J. Rotman, Galois Theory,2nd edn. (Springer, 1998) 8. S. Akiyama, Y. Tanigawa, Salem numbers and uniform distribution modulo 1. Publ. Math. Debr. 64, 329–341 (2004) 9. C. Doche, M. Mendès France, J.J. Ruch, Equidistribution modulo 1 and Salem numbers, in Functiones and Approximatio, vol. XXXIX, pp. 261–271 (2008) 10. Y. Meyer, Diophantine Quasicrystals, approximation and algebraic numbers, in Beyond Quasicrystals. Les Houches, vol. 3–16 (Springer, Berlin, 1994), p. 1995 11. Y. Meyer, Algebraic Numbers and Harmonic Analysis (North-Holland, Amsterdam 1972) 12. J. Bourgain, The Riesz-Raikov theorem for algebraic numbers. Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), 1–45, in Israel Mathematical Conference Proceedings (Weizmann, Jerusalem, 1990), p. 3 13. P. Corvaja, U. Zannier, On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France. Acta Math. 193, 175–191 (2004) 14. P. Erdös, Bernoulli, On the smoothness properties of a family of Bernoulli convolutions. Am. J. Math. 62, 180–186 (1940) 15. B. Solomyak, On the random series  ± λn (an Erdös problem). Ann. Math. 142, 611–625 (1995) 16. N. Bourbaki, Utilisation des nombres réels en topologie générale, livre III (Hermann, 1948) 17. R. Kaufman, A functional method for linear sets. Israël J. Maths. 5, 185–187 (1967)

Chapter 3

Diophantine Approximation

3.1 One-Dimensional Diophantine Approximation Throughout this chapter, [x] denotes the integral part and {x} the fractional part of the real number x so that x = [x] + {x}; moreover we shall use the notation x for the closest distance of x to an element of Z.

3.1.1 Historical Survey Every real number x ∈ R can be obtained in many ways as the limit of some sequence of rational numbers, and    1 p  ∗  . ∀q ∈ N , ∃ p ∈ Z; x −  ≤ q 2q What else? Can we have an estimate of the error, in terms of the size (complexity) of the rational: q or max(| p|, q) ? First of all, a rational number r = a/b is badly approximable by other rational numbers since             r − p  =  a − p  =  aq − bp  ≥ 1      q b q bq  bq if b, q > 0 and r = qp ; nothing better may happen: if c ≤ b1 , the inequality     0 < r − qp  < qc has no solution. This leads to the following criterion for irrationality:     Proposition 3.1.1 If the inequality 0 < x − qp  < φ(q) has infinitely many solutions, where φ > 0 is o(1/q), then x ∈ / Q. © Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_3

69

70

3 Diophantine Approximation

This criterion has been used by Fourier (1815) to prove the irrationality of the Euler basis e: 1 ∀n ∈ N, ∃An ∈ N; 0 < n!e − An = O( ); n and it is also the starting point of the proof by Apéry or Beukers of the irrationality of ζ(3) [1, 2]. The first general result in diophantine approximation is due to Lejeune-Dirichlet (1842): as a consequence of the “pigeonhole principle”, he proved the following: Proposition 3.1.2 Let x ∈ / Q; there exist infinitely many p/q such that    p 1 0 < x −  < 2 . q q (We say “every irrational number is approximable at order q 2 or simply at order 2”) This weak version of Dirichlet’s principle is easily proved: consider the real numbers 0, {x}, {2x}, . . . , {Qx} in [0, 1] that we split in Q ≥ 1 intervals of identical length; these Q + 1 numbers are distinct (α ∈ / Q), and so at least two of them fall into the same interval: |{ j x} − {kx}| = |( j − k)x − [ j x] + [kx]| ≤ 1/Q, for some 0 ≤ j, k ≤ Q. If we set q = | j − k|, 0 < q ≤ Q and q x ≤ 1/Q ≤ 1/q.

♦

At the same time, Liouville established that algebraic numbers are not very well approximated by rationals. Proposition 3.1.3 If x is an algebraic number of degree d, one can find C > 0 such that |x − qp | ≥ qCd for every rational p/q. The natural question of the converse arises immediately: is it possible to characterize this property of being algebraic in terms of approximation? A century later, the answer is definitely: No! After contributions of Thue, Siegel,..., Roth (in 1955) proved the following. Theorem 3.1.4 (Roth) If x is an algebraic number, for all ε > 0, the inequality    p 1 0 < x − q  < q 2+ε has no more than a finite number of solutions; in other words if     x can be approximated at some order b > 2 (i.e. 0 < x − qp  < q1b holds infinitely often), then x is transcendental. By the way, between q 2 and q 2+ε , a lot of place remains ! We have just to change the scale... Famous example: e is approximated at order q 2 log q/ log log q exactly. Thus the transcendence of e (proved by Hermite) cannot be a consequence of Roth’s techniques. But there exist “worse” irrational numbers as we shall see, approximated at order 2 and nothing better. Such√numbers are said to be badly approximable ones. Such is the golden number θ = 1+2 5 .

3.1 One-Dimensional Diophantine Approximation

71

3.1.2 How to Find the Best Approximations? Very lacunary q-adic expansions lead to very good approximations by truncation; this is the idea who constructed in this way transcendental decimal expan  of Liouville n! . But in the general case, one can get the Dirichlet Approximations 1/10 sions n by a suitable expansion, namely the RCF (regular continued fraction) expansion. Every rational number x = p/q can be written as a0 +

1 a1 +

,

1 a2 + · · · +

1 an−1 +

1 an

with a0 = [x] and the integers ai ≥ 1 for i ≥ 1. This is just a reinterpretation of the euclidean algorithm. Proposition 3.1.5 Every irrational number can be written, in a unique way, as the infinite continued fraction x = a0 +

1 a1 +

=: [a0 ; a1 , . . . , an , . . .]

1 a2 + · · · +

1 an−1 +

1 an + · · ·

with a0 = [x] and the integers an := an (x) ≥ 1, n ≥ 1 being called partial quotients. This expansion enjoys many properties, the most striking one being the following: for every irrational number, it provides by truncation pretty good approximations satisfying the Dirichlet principle. pn 1 pn (x) =: = a0 + , 1. (Algorithmic) Put 1 qn (x) qn a1 + 1 a2 + · · · + 1 an−1 + an if a j = a j (x). These rationals are called convergents to x and satisfy a 2-order linear recurrence relation pn+1 = an+1 pn + pn−1 , qn+1 = an+1 qn + qn−1 p0 = a0 , q0 = 1, p−1 = 1, q−1 = 0 which can be written with help of matrices: 

pn+1 pn qn+1 qn



 =

   a0 1 an+1 1 ··· . 1 0 1 0

72

3 Diophantine Approximation

Note that the pn /qn are irreducible since pn+1 qn − qn+1 pn = (−1)n for every n by computing determinants. 2. (Optimality) We have now    1 pn  1  < < x − , qn  qn+1 qn (qn + qn+1 ) qn then

   1 pn  1  < < x − . qn  an+1 qn2 (an+1 + 2) qn2

(3.1.1)

This means that we have found explicit (computable) Dirichlet’s approximations. Note that a convergent of index n approaches all the more x as the next partial quotient is big: for example, π = [3; 7, 15, 1, 292, . . .] and [3; 7, 15, 1] = 355/113 is a very good approximation to π. Proof Iterate pn+1 qn − qn+1 pn = (−1)n under the form pn 1 1 (−1)n−1 = a0 + − + ··· + qn q0 q1 q1 q2 qn−1 qn to obtain

pn = (−1)n x− qn



1 1 − + ··· qn qn+1 qn+1 qn+2



 In addition, these approximants are best possible in the strong following sense: if qn := qn (x), qn x = min{q x; q < qn+1 }, where x = d(x, Z). So we keep in mind q < qn+1 ⇒ q x ≥ qn x;

1 1 < qn x < . 2qn+1 qn+1

(3.1.2)

3. (Dynamics) The RCF expansion of a number in ]0, 1[ can be obtained  by iterating a transformation: indeed a1 (x) = x1 and an (x) = a1 (T n−1 x) where

 T , the Gauss transformation, is defined by T x = x1 . Gauss in 1832 discovdx ered an absolutely continuous measure preserved by T , μ = log1 2 1+x , referred to as the Gauss measure. Observe that μ is equivalent to the Lebesgue measure 1 m ≤ μ ≤ log1 2 m. The associated dynamical system (]0, 1[, B, μ, T ) hapm : 2 log 2 ∗ pens to be ergodic and isomorphic to a shift on (N∗ )N . We develop further this important point of view.

3.1 One-Dimensional Diophantine Approximation

73

3.1.3 Classification of Numbers We restrict ourselves to [0, 1] =: I and we aim to deduce from approximation theory a pertinent classification of real numbers in I . Definition 3.1.6 Denote by Bad the set of all badly approximate numbers x in I , i.e. those for which there exists C > 0 with     x − p  ≥ C  q  q2 for every rational p/q. There is a beautiful connection between Bad and the theory of continued fractions quoted below. Proposition 3.1.7 x ∈ Bad if and only if the sequence of its partial quotients (an )n≥1 is bounded. Proof Suppose that an ≤ K for every n; then, by (3.1.1) and (3.1.2), 1 1 ≤ < qn x, (K + 2) qn (an+1 + 2) qn and if qn ≤ q < qn+1 , q x ≥ qn x by the optimality property of the convergents; we deduce that q x ≥ C/q. Conversely,    1 pn  C  < ≤ x −  2 qn qn (an+1 ) qn2 implies an ≤ 1/C for every n.



Observe that x ∈ Bad if and only if the sequence (qn (x)) is at most lacunary. We shall invoke this property in the next Chapter. On the opposite, Liouville numbers are very well approximated by rationals and this provides a precise definition: Definition 3.1.8 Denote by L the set of Liouville numbers in I , i.e. those x ∈ I such that, for every ε > 0,     x − p  < 1  q  q 2+ε for infinitely many rationals p/q. Intermediate speeds of approximation lead to Diophantine numbers.

74

3 Diophantine Approximation

Definition 3.1.9 Fix r ≥ 1. The real number x is said to be r -Diophantine if    c p   ∃c > 0; x −  ≥ r +1 for all rationals p/q. q q The following characterizations in terms of the continued fraction expansion of x will be used later. Proposition 3.1.10 The real number x ∈ I is r -Diophantine if and only if

if and only if

∃c > 0; ∀n ≥ 0 qn+1 ≤ qnr /c,

(3.1.3)

∃c > 0; ∀n ≥ 0 an+1 ≤ qnr −1 /c.

(3.1.4)

Proof The proof goes on much the same way as above. If q r q x ≥ c for every q, then, in view of (3.1.1) and (3.1.2), c ≤ qnr qn x ≤ qnr

1 qn+1

≤ qnr −1

1 an+1

for every n ≥ 0, whence (3.1.3) and (3.1.4). Now, assuming (3.1.4), we get by (3.1.2) qn x ≥

1 C ≥ r; (an + 2)qn qn

once again, by using the optimality property, this inequality passes on to any q: fix q and n such that qn ≤ q < qn+1 ; then q x ≥ qn x ≥

C C ≥ r r qn q

and x is r -Diophantine.



Note that the set D of all Diophantine numbers is precisely the complement of L in the irrational numbers of I . Remark. It is convenient to introduce here the following robust parameter called the irrationality exponent. Definition 1 Let x be an irrational number. The irrationality exponent of x is 

 p μ(x) := sup δ > 0 : x −  ≤ q −δ infinitely often. q Clearly, μ(x) ∈ [2, ∞], μ(x) = 2 if x ∈ Bad (but not only, for example, μ(e) = 2 below) and {μ = ∞} is the set of Liouville numbers. Note the useful formula in terms of the continued fraction expansion of α:

3.1 One-Dimensional Diophantine Approximation

μ(α) = 1 + lim sup n→∞

log qn+1 (α) log an+1 (α) = 1 + lim sup · log qn (α) log qn (α) n→∞

75

(3.1.5)

We are now able to prove the precise approximation property of the base e announced previously. As we shall see, this can be deduced from the RCF expansion of e (discovered by Euler 1737) e = [2; 1  2 1 1  4 1 1  6 1 · · · 1 2n

 1 · · · ]. Theorem 3.1.11 Fix 0 < ε < 1/2; for infinitely many rational numbers p/q, |e −

p 1 log log q | < ( + ε) 2 q 2 q log q

holds, while the following inequality |e −

p 1 log log q | > ( − ε) 2 q 2 q log q

is satisfied for all but finitely many ones. Proof Looking at the RCF expansion of x := e x = [2; 1 2 1 1 4 1 . . . 1 2n 1 . . .] we see that for every n ≥ 1, a3n−2 = a3n = 1, a3n−1 = 2n; we shall thus make use of the subsequence p3n−2 /q3n−2 . Indeed, these convergents satisfy the inequalities (3.1.1):     1 1 x − p3n−2  ≤ ≤ (3.1.6)  2 2 q3n−2  a3n−1 q3n−2 (2n) q3n−2 and we are thus led to express n as a function of q3n−2 . Let us begin with a naive observation in the general case. We denote by Fn the nth Fibonacci number with F0 = F1 = 1; the following robust estimate on qn a1 · · · an ≤ qn ≤ Fn a1 · · · an ,

(3.1.7)

can be derived from the obvious one: an+1 qn ≤ qn+1 ≤ an+1 (qn + qn−1 ), inductively on n, starting with q0 = 1, q1 = a1 . Applying (3.1.7) with the sequence (an (x)), we get (since log a3 j−2 = log a3 j = 0): n n (3.1.8) log qn = log n + O(n) ∼ log n and log qn+1 ∼ log qn . 3 3 Thus, log log qn ∼ log n, and then we obtain expressing n in terms of qn :

76

3 Diophantine Approximation

n∼

3 log qn 3 log qn ∼ log n log log qn

where we have used (3.1.8). In other terms,   3 log qn log qn . n= +o log log qn log log qn

(3.1.9)

Now, ε being fixed, (3.1.9) means that 3 log qn 3 log qn (1 + ε) (1 − ε) ≤ n ≤ log log qn log log qn

(3.1.10)

for n large enough. The lower bound for 3n − 2 combined with the right inequality in (3.1.6) gives     1 log log q3n−2 1 x − p3n−2  < ≤ ( + ε) 2  2 q3n−2  (2n)q3n−2 2 q3n−2 log q3n−2 and finally (setting p = p3n−2 and q = q3n−2 ):     x − p  ≤ ( 1 + ε) log log q  q 2 q 2 log q for infinitely many p/q. This gives the first part of the theorem. In the reverse direction, let ε > 0. Start from q ≥ 1 and let n = n(q) be the integer such that qn ≤ q < qn+1 . Since an+1 ≤ 23 (n + 1), we get for large q, with help of (3.1.2) and (3.1.10):     1 1 1 1 3 3 −ε −ε q x ≥ qn x ≥ ≥ ≥ ≥ qn+1 + qn 2 nqn 2 nq (an+1 + 2) qn       1 3 log log q 1 log log q 3 −ε −ε ≥ − 2ε , ≥ 2 q 2 q log q 2 q log q whence the proof of the second part. 

3.1.4 First Arithmetical Results We have seen that the irrational number e is not Liouville, not even diophantine, by using its RCF expansion, but this is of no help when we consider more generally er , r ∈ R. We begin this subsection with recalling usual notations and basic facts on Padé approximants (see [3, 4] for complements) in order to prove the following:

3.1 One-Dimensional Diophantine Approximation

77

Theorem 3.1.12 For every r ∈ Q, er is an irrational number. Let f (z) be a power series in one variable with rational coefficients f (z) =



ck z k , ck ∈ Q.

k≥0

Definition 3.1.13 For non-negative integers n, m, the Padé approximant [m/n] f of f at z = 0 is any rational fraction A/B in Q[[z]] such that deg A ≤ m, deg B ≤ n and   B(z) f (z) − A(z) = Oz=0 z n+m+1 . When the pair (A, B) exists, it is not necessarily unique but the fraction A/B must be. Now the existence of such a pair relies on a linear recurrence system and on the following Hankel determinants: Definition 3.1.14 For k ≥ 1, the Hankel determinant Hk ( f ) of order k associated to f is the following one : ⎤ c0 c1 · · · ck−1 ⎢ c1 c2 · · · ck ⎥ ⎥ ⎢ ⎢ . . . . ⎥ ⎥ ⎢ Hk ( f ) = ⎢ . ⎥ ⎥ ⎢ . . . ⎣ . . . . ⎦ ck−1 ck · · · c2k−2 ⎡

If Hn ( f ) is non-zero, then the Padé approximant [((n − 1)/n] f exists and we have Hn+1 ( f ) 2n z + Oz=0 (z 2n+1 ). f (z) − [(n − 1)/n] f (z) = Hn ( f ) Finally, one will easily be convinced that all the Padé approximants can be deduced from the main one [(n − 1)/n] f . The famous case of the exponential f (z) = e z has been exploited by Hermite [5]; here the Hankel determinants, thus the approximants, can easily be computed. We prefer a different approach due to Siegel [6] consisting in a direct determination of the approximant pair [m/n] f =: (Pm,n , Q m,n ) by derivation. Let us denote by D the derivation operator, noting that D(Q.e z ) = e z · (I + D)(Q), and by induction,   D k Q.e z = e z · (I + D)k (Q) , for every polynomial Q. Now, the polynomials Pm,n , Q m,n , of degree ≤ m and ≤ n, respectively, are assumed to satisfy   Rm,n (z) := Q m,n (z) e z − Pm,n (z) = Oz=0 z n+m+1 . A first property, specific to the exponential function, is the following:

(3.1.11)

78

3 Diophantine Approximation

Lemma 3.1.15 If Q n,m (0) = 0, then we have Pm,n (−z) =

Q m,n (0) Q n,m (z). Q n,m (0)

Proof From (3.1.11) we get Q m,n (z) − e−z Pm,n (z) = Oz=0 (z n+m+1 ) which can be written:   Pm,n (−z) e z − Q m,n (−z) = Oz=0 z n+m+1 ; by unicity of such a pair up to a multiplicative constant, we must have Pm,n (z) =  λQ n,m (−z), whence Q m,n (0) = Pm,n (0) = λQ n,m (0) and the lemma. We get from the definition (3.1.11) D m+1 Rm,n = e z · (I + D)m+1 Q m,n =: e z Sm,n .

(3.1.12)

The degree of the polynomial Sm,n is ≤ n but, since Sm,n = e−z · D m+1 Rm,n , Sm,n is also a formal series of order ≥ n at z = 0 by (3.1.11); it follows that Sm,n (z) = (−1)n z n with a suitable normalization. Now, Eq. (3.1.12), written as D m+1 Rm,n (z) = (−1)n z n e z , can be reversed with help of the integral remainder Taylor formula; in view of (3.1.11) once more, one gets  (−1)n z n t t e (z − t)m dt m! 0  (−1)m+n z n+m+1 1 n = t (t − 1)m e zt dt m! 0

Rm,n (z) =

with prescribed order n + m + 1 at z = 0. From (I + D)m+1 Q m,n = (−1)n z n now, we derive formally Q m,n (z) = (−1)n (I + D)−(m+1) z n = (−1)n (I − (m + 1)D + · · · + (−1)n =

n k=0

(−1)n−k

that is Q m,n (z) =

(m + k)! k  n  D z , m!k!

(m + 1) · · · (m + n) n  n  D ) z n!

n  n!(n + m − j)! (−1) j z j ∈ Z[x]. j!m!(n − j)! j=0

(3.1.13)

Invoking Lemma 3.1.15 with λ = n!/m! as easily seen, we finally obtain Pm,n (z) =

m  (n + m − j)! j z ∈ Z[x]. j!(m − j)! j=0

(3.1.14)

3.1 One-Dimensional Diophantine Approximation



If we substitute

∞

(n + m − j)! =

79

e−t t n+m− j dt

0

in each sum (3.1.13) and (3.1.14), we get an integral representation for these polynomials:  ∞ 1 e−t (t − z)n t m dt Q m,n (z) = m! 0  ∞ 1 Pm,n (z) = e−t (t + z)m t n dt. m! 0 The initial requirement (3.1.11) on (Pm,n , Q m,n ) can be checked on these expressions if we observe, with a last change of variable, that (for Re z > 0)  z n+m+1 e z 1 −zt m n e t (t − 1) dt. m! 0  z m+n+1 ∞ −zt m n Q m,n (z) = e t (t − 1) dt m! 1 Rm,n (z) =

and

z m+n+1 e z Pm,n (z) = m!



∞

e−zt t m (t − 1) dt n

(3.1.15) (3.1.16)

(3.1.17)

1

Proof of theorem 3.1.12 : The theorem follows easily from the diagonal case: (3.1.15), (3.1.13), (3.1.14) with m = n. Assume that a ∈ Z∗ , so that Q n (a) and Pn (a) are integers. Clearly, Rn (a) = 0, and since t (1 − t) ≤ 1/4 on [0, 1],  2n / Q in view of |Rn (a)| ≤ n!a a2 max(1, ea ) → 0 as n → ∞. This proves that ea ∈  proposition 3.1.1. The case of er , r ∈ Q∗ , follows.

3.2 The Gauss Ergodic System The Gauss measure μ is easily checked to be T -invariant: μ(T −1 A) = μ(A) for every A ∈ B; actually, it is sufficient to test this identity on the sub-intervals of ]0, 1[; note 1 1 , a+n [ is a disjoint union of intervals so that that T −1 ]a, b[= ∪n≥1 ] b+n 

μ T

−1

  ]a, b[ = μ ] 

n≥1

  1 1 1 1  a+n d x , [ = ; 1 b+n a+n log 2 n≥1 b+n 1+x

now 

1 a+n 1 b+n

dx = log(a + n + 1) − log(a + n) − log(b + n + 1) + log(b + n) 1+x

80

3 Diophantine Approximation

and     a+n 1  a+n+1 − log log log 2 n≥1 b+n+1 b+n   1 b+1 = μ (]a, b[) . = log log 2 a+1

  μ T −1 ]a, b[ =

It is less obvious to exhibit this invariant measure ex nihilo: actually it is the unique absolutely continuous T -invariant probability measure, and it can be obtained as a fixed point of some operator [7]. The Gauss dynamical system (]0, 1[, B, μ, T ) enjoys a mixing property that means a kind of independence for the variables an (x), and which turns out to be very powerful. If ω = a1 · · · an is a word of length |ω| = n on the alphabet N∗ , the cylinder [ω] is the following subset of ]0, 1[: [ω] = {x ∈ ]0, 1[ , ak (x) = ak , 1 ≤ k ≤ n.} If we denote by pn /qn , the rational number associated to ω by pn = qn

1 a1 +

= [0; a1 , . . . , an ],

1 a2 + · · · +

1 an−1 +

1 an

the cylinder in fact is the interval  pn pn + pn−1 , [ω] = , qn qn + qn−1 

of length 1/qn (qn + qn−1 ). Distinct words lead to disjoint cylinders and the σ-field Fn generated by the cylinders of size n consists in a (at most) countable union of such sets; moreover   (3.2.1) Fn = B. σ n

Lemma 3.2.1 There exist positive constants c, C such that, for every n ≥ 0, for every A ∈ Fn and B ∈ B, cμ ( A) μ (B) ≤ μ(A ∩ T −n B) ≤ Cμ ( A) μ (B) . As a consequence,

(3.2.2)

3.2 The Gauss Ergodic System

81

Corollary 3.2.2 The Gauss system (]0, 1[, B, μ, T ) is ergodic and even strong mixing. Proof Since μ is equivalent to the Lebesgue measure, we are reduced to establish an inequality similar to (3.2.2) with m instead of μ. Let us fix n ≥ 0, A ∈ Fn and I = [s, t[. We claim that 1 m(A)m(I ) ≤ m(A ∩ T −n I ) ≤ 2m(A)m(I ). 2

(3.2.3)

The integer n being fixed, T n admits infinitely many inverse branches, coded by the different words of length n, which we denote by χω , |ω| = n. When ω = a1 · · · an , it is clear that χω is the Möbius function χω (x) = [0; a1 , . . . , an + x] =

pn + x pn−1 , x ∈ [0, 1[. qn + xqn−1

(3.2.4)

Each χω is monotonic on [0, 1[ (increasing or decreasing, according to the parity of n) and χω ([0, 1[) = [ω]. We now pay attention to the intersection: [ω] ∩ T −n [s, t[ with ω = a1 · · · an ; clearly z ∈ [ω] ∩ T −n [s, t[ if and only if z = χω (x) with s ≤ x < t, thus, with help of (3.2.4), m([ω] ∩ T −n I ) = |χω (t) − χω (s)| = t −s 2qn (qn + qn−1 ) 1 = m(I )m([ω]). 2

t −s (qn + sqn−1 )(qn + tqn−1 )

≥

In the opposite direction, observe that 2 (qn + sqn−1 ) (qn + tqn−1 ) ≥ 2qn2 ≥ qn (qn + qn−1 ) which finally provides 1 m(I )m([ω]) ≤ m([ω]) ∩ T −n I ) ≤ 2m(I )m([ω]) 2 for every interval I and |ω| = n; using σ-additivity, we get right after 1 m(I )m(A) ≤ m(A ∩ T −n I ) ≤ 2m(I )m(A) 2 for every interval I and A ∈ Fn . Let us fix now n and A ∈ Fn , and consider 1 B A = {B ∈ B, m(B)m(A) ≤ m(A ∩ T −n B) ≤ 2m(A)m(B)}. 2

82

3 Diophantine Approximation

We have just seen that B A contains all the intervals in [0, 1[; since it also contains any finite disjoint union of intervals, it must contain B by the monotone class theorem. Whence the claim (3.2.3) and the lemma.  The ergodicity of the system follows. Suppose that B is an invariant Borel set; thus, from (3.2.2), μ(A ∩ B) ≥ cμ(A)μ(B) for every A ∈ F = ∪n Fn , B ∈ B∞ and by the same arguments, μ(A ∩ B) ≥ cμ(A)μ(B) holds for every A ∈ B in view of (3.2.1); taking A = B c , we get μ(B c )μ(B) = 0, and μ(B) = 0 or 1. Actually a much stronger property holds for this system: if Bn is the σ-field σ{ak , k ≥ n} and B∞ = limn ↓ Bn is the tail σ-field, every B ∈ B∞ (even not invariant) must be trivial. Indeed, if B ∈ B∞ , for any n one can find E n ∈ B such that B = T −n E n so that, by the lemma, cμ(A)μ(B) = cμ(A)μ(E n ) ≤ μ(A ∩ T −n E n ) = μ(A ∩ B) whenever A ∈ Fn . As before, we conclude that μ(A ∩ B) ≥ cμ(A)μ(B) holds for every A ∈ B and finally that μ(B) = 0 or 1. This stronger property is expressed as: T is a Kolmogorov shift. It implies that the system is strong mixing. Indeed, we have to show that lim μ(A ∩ T −n B) = μ(A)μ(B), for every A, B ∈ B.

n→∞

(3.2.5)

Clearly, T −n B ∈ Bn for every n hence, by definition of the conditional expectation, μ(A ∩ T −n B) =



 T −n B

1 A dμ =

T −n B

E(1 A /Bn ) dμ;

we write now          μ(A ∩ T −n B)−μ(A)μ(B) =    −n (E(1 A /Bn ) − μ(A)) dμ  T B       ≤ E(1 A /Bn ) − μ(A) dμ But a classical result in Hilbert spaces asserts that, for a decreasing sequence of closed convex subsets (Cn ) with non-empty intersection C (here subspaces!), the projections PCn converge pointwise to the projection PC [8]. This applies to L 2 (X ), the subspaces Cn = L 2 (Bn ), C = L 2 (B∞ ) and the corresponding projections Pn := E(·/Bn ) and P := E(·/B∞ ), implying that E( f /Bn ) → E( f /B∞ ) in L 2 (X ) for every f ∈ L 2 (X ). Choosing f = E(1 A /B∞ ), we have E( f /Bn ) =  E(1 A /Bn ) but also f = μ(A) a.e. since B∞ is trivial (this is the main point). Finally |E(1 A /Bn ) − μ(A)| dμ tends to 0 and the mixing property is established.  The Birkhoff ergodic theorem (cf. Chap. 2) tells us that for f ∈ L 1 ([0, 1]),  1  n f (T x) → f dμ μ − a.e. N n u n }, n ≥ 1. As usual (Borel–Cantelli lemma), we have that m(lim sup An ) = lim ↓ m(∪n≥ p An ) ≤ lim p

n

p



m(An ) = 0

n≥ p

 1 since m(An ) = m({x ∈ I ; a1 (x) > u n }) < 1/u n and < ∞. un ∞ 1 c • If now 1 u n = ∞, we shall show that μ(lim inf n An ) = 0 as in Borel– Cantelli’s lemma, with inequality (3.2.2) standing in for independence. For p ≤ n, Acp ∩ · · · ∩ Acn = {x ∈ I ; a p (x) ≤ u p , . . . , an (x) ≤ u n }  {x ∈ I ; a p (x) = v p , . . . , an (x) = vn } = 1≤vk ≤u k p≤k≤n

=

 v1 ≥1,...,v p−1 ≥1 v p ≤u p ,...,vn ≤u n

[v1 · · · vn ]

3.2 The Gauss Ergodic System

85

with the previous notations. For any such cylinder [ω] = [v1 · · · vn ] and any Borel set B, we recall that 1 m([ω] ∩ T −n B) ≥ m([ω])m(B); 2 thus, taking B = {x ∈ I ; a1 (x) > u n+1 } = [0, 1/u n+1 ) so that T −n B = An+1 , and summing over the words [v1 · · · vn ] involved, we get m(Acp ∩ · · · ∩ Acn ∩ An+1 ) ≥

1 1 m(Acp ∩ · · · ∩ Acn ) . 2 u n+1

Let us fix p ≥ 1, and denote by sn the number m(Acp ∩ · · · ∩ Acn ), we see that m(Acp ∩ · · · ∩ Acn ∩ An+1 ) = sn − sn+1 ≥

sn 2u n+1

then sn+1 ≤ (1 − ≤e

−

n+1 1 1 )sn ≤ (1 − ) j= p 2u n+1 2u j

n+1

1 j= p 2u j

.

 1 But, by our assumption on (u n ), ∞ j= p u j = ∞ for every p, which implies that  m(∩ j≥ p Acj ) = 0 for every p and m(∪ p ∩ j≥ p Acj )) = 0 in turn.

3.3 Back to Transcendence 3.3.1 Metric Results It is clear from Proposition 3.1.7 that Bad is uncountable and thus must contain transcendental numbers. But a fundamental result, already observed, is that Bad is a set of zero Lebesgue measure. Thanks to Roth’s theorem, new transcendental numbers have appeared including Liouville’s numbers L: the 2 + ε-approximable numbers for some ε > 0, which we denote by R. But once more we are far from getting this way all the transcendental numbers since m(R) = 0; this is a consequence of a result of Khintchine that follows. Let φ : N → R+ be some non-decreasing function. Definition 3.3.1 Kφ is the set of all x ∈ [0, 1] approximable at order 1/q 2 φ(q), i.e. Kφ = {x/q x < 1/qφ(q) infinitely often}

86

3 Diophantine Approximation

The following theorem, which is also a Borel–Cantelli-type theorem, with the difficulty that there is no evident independence of the variables, gives the size of these sets. Theorem 3.3.2 (Khintchine 1924) The Lebesgue measure of Kφ is 

 0 if 1/qφ(q) < ∞ q≥1 1 if q≥1 1/qφ(q) = ∞

Proof The proof follows from the Borel–Bernstein theorem, the Lévy theorem and the approximation property of the convergents (3.1.1). Once more, the first case is much more elementary and left to the reader. We focus on the second one, when  1/qφ(q) =∞. q≥1 If we are able to prove that, for almost every x ∈ I , an+1 (x) ≥ φ(qn (x)) infinitely often, we simply deduce from (3.1.1):     1 1 x − pn (x)  < ≤ infinitely often  qn (x)  an+1 (x)qn2 (x) φ(qn (x))qn2 (x)   for such x’s and thus m Kφ = 1. Let us call H the Lévy constant, H = π 2 /12 log 2; if the integer N is such that N > e H , then, for almost all x ∈ I , qn (x) < N n for n ≥ n(x). Since φ is non-decreasing,  N n ≤q 0, so that R in turn is a negligible set. Thus, most of transcendental numbers lie in Rc ; this is the case for the base e since e − 2 belongs to Kφ with φ(q) = log q/ log log q.

3.3 Back to Transcendence

87

However, Diophantine approximation has been used by Hermite for proving the transcendence of e, BUT, actually, simultaneous approximation.

3.3.2 Simultaneous Diophantine Approximations A very easy generalization of the irrationality criterion (Proposition 3.1.1) gives a process for proving transcendence. Lemma 3.3.3 Let x ∈ R∗ be such that the following holds: For all integers r ≥ 0, a0 , . . . , ar with a0 = 0, one can find r + 1 integral sequences (qn ), ( p1,n ), . . . , ( pr,n ) such that  1. a0 qn +  rk=1 ak pk,n =  0 for infinitely many n ≥ 1, 2. limn→∞ qn x k − pk,n = 0 for 1 ≤ k ≤ r . Then x is transcendental. This statement means that transcendence can, therefore, be established by producing good simultaneous approximants (i.e. approximants with the same denominator qn ) to consecutive powers of x. This idea has already been used by Liouville, in proving that e is not a quadratic irrational number and it leads to Hermite’s method for transcendence.  Proof The proof is straightforward. Assume that rk=0 ak x k = 0 for some integers a0 , . . . , ar , with a0 = 0 (otherwise, we can lower r ). Then, for every n ≥ 1, a0 q n +

r  k=1

and

ak q n x k −

r  k=1

ak pk,n +

r 

ak pk,n = 0,

k=1

r r     k     a = q a x − p q + ak pk,n  ≥ 1  0 n  k n k,n  k=1

k=1

 since the integer a0 qn + rk=1 ak pk,n is non-vanishing by Hypothesis 1. Hypothesis 2. leads to a contradiction by letting n tend to infinity in the LHS.  We now present a proof of the transcendency of e, inspired by simultaneous Padé approximants, which are usually called: Hermite–Padé approximants. The following theorem provides a generalization of the Hermite formulas (3.1.15), (3.1.16), (3.1.17). Theorem 3.3.4 Given n 1 , . . . , n r with n 1 + · · · + n r = n, and m ≥ 0, one can find polynomials Q of degree ≤ n and P1 , . . . , Pr of degree m k := n + m − n k respectively, k = 1, . . . , r , such that   Rk (z) = Q (z) ekz − Pk (z) = Oz=0 z n+m+1

(3.3.1)

88

3 Diophantine Approximation

Moreover this solution normalized by Q(0) = 1 admits the following integral representation for Re z > 0:  ∞ z n+m+1 Q (z) = T (x)e−zx d x, (n + m)! 0  z n+m+1 ekz ∞ T (x) e−zx d x, Pk (z) = (n + m)! k  z n+m+1 ekz k Rk (z) = T (x) e−zx d x, (n + m)! 0 where T (x) = x m

r j=1

(x − j)n j .

Proof The existence and uniqueness (up to a constant) of the solution to (3.3.1) rest on considerations on the underlying linear system. We attempt to give some heuristic justification for the description of the normalized solution. Taking r = 1 goes back to the classical approximants and identities (3.1.15), (3.1.16), (3.1.17). Looking forward, a similar integral form for Q, Pk , Rk , it becomes clear, after the remark below, that it must be the one in the statement. Lemma 3.3.5 Let ϕ ∈ C N [z], M ≤ N and k be any non-negative integer. Then Ik (z) := ekz z N +1



∞

e−zt ϕ (t) dt (Re z > 0)

k

is a polynomial of degree at most N − M if and only if (t − k) M divides ϕ. If ϕ has integral coefficients, Ik has integral coefficients divisible by M!. Finally, if k = 0 and ϕ(t) = M≤ j≤N c j t j , then I0 (z) = M!c M z N −M + (M + 1)!c M+1 z N −M−1 + · · ·

(3.3.2)

This imposes to choose ϕ as T and N as n + m + 1, whence the description of Q, Pk , Rk in the Theorem. The lemma is a variation around the classical identity: 

∞

e−zt t p dt =

0

p! (Re z > 0) ; z p+1

 N +1 ∞

−zt since Ik (z) = z ϕ (t + k) dt, and since M! divides the coefficients of 0 e ϕ(t) as soon as it divides those of ϕ(t + k), we can assume k = 0 without loss of generality. We get

z N +1



∞ 0

e−zt ϕ (t) dt =

 0≤ j≤N

j!c j z N − j

3.3 Back to Transcendence

89

 if ϕ (t) = 0≤ j≤N c j t j , and this new polynomial has degree at most N − M if and only if c j = 0 for j = 0, 1, . . . , M − 1, and coefficients divisible by M! if c j ∈ Z. This makes (3.3.2) quite clear.  The power of such representations holds in that we get in fact more than the expected property (3.3.1): we are able to estimate the errors Rk (z) for any z (not only when z is close to 0). By specializing the parameters, we can deduce the transcendency of er for every r ∈ Q. Theorem 3.3.6 (Hermite) The number e is a transcendental number. Proof We just apply the previous theorem with n k = n for every k, m = n − 1 and z = 1, or the previous lemma with N = nr + n − 1 and ϕ = Tn defined below. Also another normalization will be needed; we are thus led to consider  ∞ 1 Tn (x) e−x d x, qn = (n − 1)! 0  ∞ ek Tn (x)e−x d x pk,n = (n − 1)! k and εk,n = qn e − pk,n k

where Tn (x) = x

r n−1

j=1 (x

ek = (n − 1)!



k

Tn (x)e−x d x,

0

− j)n . Clearly, for every k = 0, . . . , r , εk,n → 0 as n → ∞.

But we have to check that the qn and pk,n are integers, then, that condition 1. in Lemma 3.3.3 is satisfied for infinitely many of them. The first point is a consequence of Lemma 3.3.5; indeed, observe that Tn (x) = cn−1 x n−1 + cn x n + · · · with ck ∈ Z, and  ∞ Q n (1) qn = with Q n (x) = x N +1 Tn (t)e−t dt. (n − 1)! 0 Now, the relation (3.3.2) with M = n − 1 gives qn ∈ Z and more precisely qn = cn−1 + ncn + · · · with cn−1 = [(−1)r r !]n . Similarly, we have (using Lemma 3.3.5 with M = n): pk,n

Pk (1) = with Pk (x) = x N +1 ekx (n − 1)!

 k

∞

Tn (t)e−t dt,

(3.3.3)

90

3 Diophantine Approximation

which even implies that all the pk,n are divisible by n. Now, let a0 , . . . , ar be integers as in the assumptions of Lemma 3.3.3 and take for n, a prime number which is both > r and > |a0 |. This infers that condition 1. in Lemma 3.3.3 must hold for such n, since a0 qn is not divisible by n in view of (3.3.3) while rk=1 ak pk,n is. And the proof is complete.  Comments 1 1. The zeta function at even integers can always be written in terms of powers of π 2 22k−1 Bk 2k π ζ(2k) = (2k)! where the Bernoulli numbers Bk are rational while the π 2k are not! This implies that ζ(2k) ∈ / Q, even is transcendental. The value of the zeta function at odd integers raises still open questions. The initial proof by Apéry [1] of the irrationality of ζ(3) involved rational approximations which have been shown later to be certain Hermite–Padé approximants [2, 9, 10]. Many attempts to tackle ζ(5) by this method unfortunately failed. We are left with the following result (Zudilin [11]): at least one among ζ(5), ζ(7), ζ(9), ζ(11) is irrational. 2. A famous conjecture in quantitative simultaneous approximation is due to Littlewood and is still open despite many recent progress. It asserts that lim inf qqαqβ = 0 for all real numbers α, β, q→∞

and cannot be derived from the multidimensional pigeonhole principle, (see [12–14] for a survey). 3. Kronecker’s theorem deals with simultaneous and non-homogenous approximation as already pointed out. Quantitative versions of this theorem should have interesting applications (see [15]).

3.4 Exercises 1. (Fibonacci) Prove that gcd(Fn , Fm ) = Fgcd(n,m) by using the matrix identity: 

Fn+1 Fn Fn Fn−1



 =

11 10

n ,

and suggest a generalization to some recurrent sequences of this divisibility property. 2. Liouville has proved that e is not a quadratic irrational number in the following way. (i) For every n ≥ 1, there exist explicit An and Cn in N such that     1 1 1 , 0 < n! − Cn = O . 0 < n!e − An = O n e n

3.4 Exercises

91

(ii) Now suppose that ae2 + be + c = 0 for some (a, b, c) = (0, 0, 0); prove that   1 ∀n, dn := a An + bn! + cCn = O n and get a contradiction. 3. Find the best diophantine approximations for the real number tanh 1 =

e2 − 1 = [0; 1 3 5 . . . 2n − 1 . . .]. e2 + 1 1

4. Let k ≥ 2 be an integer, and let x = e k . It is known that we have x = [a0 ; a1 , · · · , an , · · · ] with a3n−3 = a3n − 1 and a3n−2 = (2n − 1)k − 1, n ≥ 1. Using the techniques of this Chapter, show that   x − p  ≤ ( 1 + ε) log log q q 2k q 2 log q for infinitely many rational numbers

p q

and

  x − p  ≥ ( 1 − ε) log log q q 2k q 2 log q for all but a finite number of rational numbers qp . 5. i) By using Padé approximants to e x and formulas (3.1.14), (3.1.15) and (3.1.16), prove that (−1)n n!n! √ e(1 + o(1)) q n e − pn = (2n + 1)! where pn = Pn (1), qn = Q n (1). ii) Prove that qn =

(2n)! √ (1 + o(1)). n! e

iii) Recover this way that e − 2 ∈ Kϕ with ϕ(q) = log q/ log log q. 1 ∞ m!n! .) (Remind that k! = 0 e−t t k dt and 0 l m (1 − t)n dt = (m+n+1)! 6. Prove that π is an irrational number by using Padé approximants to e z (the transcendency has been proved by Lindemann).

92

3 Diophantine Approximation

7. Could you identify the limit of the harmonic mean 1 a1 (x)

n + ··· +

1 an (x)

as n → ∞, for almost every x ∈ [0, 1)? 8. (Coboundary equation) For α ∈ / Q, we consider the coboundary equation (∗) f (x + α) − f (x) = g(x),

f, g 1-periodic.

Here, g ∈ C ∞ (R) is fixed, and one looks for solutions f ∈ L 1 (T). i) If α is an algebraic number, show that f ∈ L 1 (T) exists and indeed f itself of f and g). belongs to C ∞ (compare the Fourier coefficients  ii) Let now (cn ) ∈ 2 \1 and put g(x) = n≥0 cn (e2iπqn α − 1)e2iπqn x where (qn = qn (α)) are the denominators of α; prove that α can be chosen in such a way that qn α ≤ e−qn for every n ≥ 0 and that the associated g belongs to C ∞ . In that case, prove that equation (∗) admits a solution in L 2 (indeed in L p for every p < ∞) but no bounded one (Hint: any solution has a lacunary spectrum). 9. (Legendre’s theorem) Let x ∈ / Q and a rational qp both in [0, 1), such that       p p p x − q  < 2q1 2 ; we shall prove that q is a convergent of x. Let us write x − q  = qθ2 , with 0 < θ < 21 .   p (i) If q jj

1≤ j≤n pn +ω pn−1 qn +ωqn−1

are the convergents of

p q

= [0; a1 , a2 , . . . , an ], we define ω by

the relation = x; show that 0 < ω < 1. (ii) What is the continued fraction expansion of x? Observe that qp is one of its convergents. 10. Deduce from the previous exercise that with the notations of Exercise 5., Pn (1)/Q n (1) are convergents to e. 11. (Another description of Bad) Remind that x ∈ Bad if there exists C > 0 such that q x > Cq for all integers q > 0. We now define F = { f : [1, ∞[→]0, ∞[, f non-increasing} and we claim that : if x ∈ Bad then the inequality q x ≤ f (q)  with f ∈ F and f (q) < ∞ has at most a finite number of solutions. (i) Let x ∈ [0, 1] be an irrational number and suppose that x ∈ / Bad; select integers 1 < q1 < q2 < ... < qn < ... such that q j x ≤

1 2jq

∀ j ≥ 1; j

3.4 Exercises

93

(ii) Establish the first implication (⇐=) by considering the function f (t) =

 e −t e q j , t ≥ 1. jq 2 j j≥1

(iii) Let now x ∈ Bad, f ∈ F, and 1 ≤ q1 < q2 < . . . < qn < . . . be such that q j x ≤ f (q j ) ∀ j ≥ 1.  Prove that f (q) = ∞ by bounding the partial sums from below. Deduce from this the second implication (=⇒). (iv) Is this result coherent with Khintchine’s theorem ? 12. (Lévy’s constant) The Lagrange theorem says that the fraction continued fraction expansion of a quadratic irrational number is ultimately periodic. The real number x ∈ [0, 1) is said to have a Lévy constant if limn→∞ n1 log qn (x) exists and we write β(x) for the limit. We intend to describe a class of real numbers with this property. (i) Let x, y ∈ [0, 1) verifying T k x = T  y for some integers k,  ≥ 0. If  = k + j, j ≥ 0, one can exhibit two integers a, b such that qn+ j (y) = aqn (x) + bpn (x), n ≥ 0 (Mn+ j (y) and Mn (x) have to be compared). Conclude that y admits a Lévy constant as soon as x does and in this case, β(y) = β(x). (ii) Let x ∈ [0, 1) be of the form, x = [0; a1 , . . . , ak ]. Show that β(x) exists and that k−1 1 log T j x. β(x) = − k j=0 With help of Lagrange’s theorem, conclude that any quadratic irrational number admits a Lévy constant. 13. We determine in this exercise the diagonal Padé approximants to the log function near x = 1. Let Pn and Q n be two polynomials of degree n satisfying  Q n (x) log x − Pn (x) = Ox=1 (1 − x)2n+1 . (1) If Q n (x) =

n

0 tjx

j

and tn = 1, prove by using a differentiation method that

Q n (x) =

n  2  n k=0

k

 x ,

(Hint : x n+1 [Q n (x) log x](n+1) = (−1)n n!

Q n (1) =

k

n

j=0 (−1)

2n n

j j ⎛t j x ⎞

n ⎝ ⎠ j

 . = n!(x − 1)n .)

94

3 Diophantine Approximation

(2) Observe that Q n ∈ Z[x], and for x ≥ 0, establish the following: √ √ (1 + x)2n ≤ Q n (x) ≤ (1 + x)2n . 2(n + 1) (3) Show that dn Pn ∈ Z[x] where dn = LC M(1, ..., n). (4) We are left with the estimate of the error term E n (x) = log x −

Pn (x) . Q n (x)

Prove that we have exactly x Q 2n (x)E n (x) = (1 − x)2n , and together with 2) deduce bounds for the term E n (x) on the compact interval [1 − ε, 1 + ε], 0 < ε < 1.

References 1. R. Apéry, Irrationalité de ζ(2) et ζ(3). Astérisque 61, 11–13 (1979) 2. F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11, 268–272 (1979) 3. C. Brezinski, Padé-type approximation and general orthogonal polynomials (Birkhäuser Verlag, 1980) 4. E.M. Nikishin, V.N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs, 92 (American Mathematical Society, Providence, RI, 1991) 5. C. Hermite, J. Reine Angew. Math. 79, 324–338 (1874) 6. A.B. Shidlovskii, Transcendental Numbers, de Gruyter Studies in Mathematics, 12 (Walter de Gruyter & Co., Berlin, 1989) 7. A. Broise, Transformations dilatantes de l’intervalle et théorèmes limites. Études spectrales d’opérateurs de transfert et applications. Astérisque 238, 1–109 (1996) 8. F. Hirsch, G. Lacombe, Elements of Functional Analysis, Graduate Texts in Mathematics, 192 (Springer, New York, 1999) 9. W. Van Assche, Hermite-Padé rational approximation to irrational numbers. Comput. Methods Funct. Theory 10, 585–602 (2010) 10. M. Prévost, A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants. J. Comput. Appl. Math. 67, 219–235 (1996) 11. V.V. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. (Russian) Uspekhi Mat. Nauk 56 (2001), (340), 149–150; translation in Russian Math. Surveys 56, 774–776 (2001) 12. M. Einsiedler, A. Katok, E. Lindenstrauss, Invariant measures and the set of exceptions to the Littlewood conjecture. Ann. Math. 164, 513–560 (2006) 13. M. Queffélec, An introduction to Littlewood’s conjecture, Dynamical systems and Diophantine approximation, 127–150, Sémin. Congr., 19, Soc. Math. France, Paris (2009) 14. A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood Conjecture. Bull. A. M. S. 45, 117–134 (2008) 15. M. Weber, On localization in Kronecker’s Diophantine theorem. Unif. Distrib. Theory 4, 97– 116 (2009)

Chapter 4

General Properties of Dirichlet Series

4.1 Introduction Notation: For a real number θ, we denote by Cθ the following vertical half-plane: Cθ = {s ∈ C; Re s > θ}. A general Dirichlet series is a series of the form ∞ 

an e−λn s

(4.1.1)

n=1

where an ∈ C, (λn ) is an increasing sequence of non-negative real numbers, tending to +∞ and s ∈ C. Note that the case λn = n corresponds to power series with the change of variable z = e−s . The basic general fact concerning such series is the following. Lemma 4.1.1 (Jensen’s lemma) Suppose that the series (4.1.1) converges at some point s0 . Then, it converges for each s such that Re s > Re s0 , with uniform convergence on each cone   1 |s − s0 | ≤ , Sϕ = s; Re(s − s0 ) cos ϕ where 0 ≤ ϕ < π2 .

 Proof We can assume that s0 = 0, i.e. that ∞ n=1 an converges. The main point is the following fact. Assume that the complex sequence (μn ) is of bounded variation, namely, ∞  |μn+1 − μn | + |μ1 | ≤ C < ∞. (4.1.2) n=1

© Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_4

95

96

4 General Properties of Dirichlet Series

 Then, the series  an μn are all convergent, with uniform convergence for fixed C. Indeed, set rn = ∞ j=n a j . A first Abel summation shows that N 

an μn =

n=1

N 

rn (μn − μn−1 ) + r1 μ1 − r N +1 μ N ,

n=2

  showing that the series an μn converges, since the series rn (μn − μn−1 ) is absolutely convergent and |r N +1 μ N | ≤ C|r N +1 |. Now, a second Abel summation shows that ∞ ∞   an μn = rn (μn − μn−1 ) + r N μ N , n=N

n=N +1

so that setting ρ N = supn≥N |rn |, we have |

∞ 

an μn | ≤ ρ N (

∞ 

|μn − μn−1 | + |μ N |) ≤ 2ρ N C,

N +1

n=N

proving the claimed uniform convergence. For the special case μn = e−λns , it remains to observe that (setting s = σ + it, using that (λn ) increases and that |μ1 | ≤ 1 if Re s > 0)  μn − μn+1 =

λn+1

λn

se−st dt, whence

∞ 

 |μn+1 − μn | ≤ |s|

n=1

∞

e−σt dt

λ1

1 |s| ≤ if s ∈ Sϕ . ≤ σ cos ϕ



In view of Lemma 4.1.1, we can attach to each Dirichlet series (4.1.1) a number σc ≥ −∞, called the convergence abscissa, which is the analog of the radius of convergence for a power series, and verifies Re s > σc =⇒ (4.1.1) converges; Re s < σc =⇒ (4.1.1) diverges. Still nowadays, a very good reference for the theory of general Dirichlet series is the book of Hardy and Riesz [1]. But the most interesting case (to which we shall stick), in connection with multiplicative number theory, from which those series were born, is the case of ordinary Dirichlet series, for which λn = log n, and for which a very good modern reference is the book of Helson [2]. Contrary to the case of power series, several abscissas will be attached to (4.1.1). We elaborate on this in the next subsection. Also observe that the “simplest” ordinary Dirichlet series, analogous to ∞  n −s z = 1/(1 − z), is the series ζ(s) = the power series ∞ n=0 n=1 n , the famous zeta function!

4.2 Convergence Abscissas

97

4.2 Convergence Abscissas 4.2.1 The Bohr–Cahen Formulas  −s For a Dirichlet series ∞ always supposed to diverge at the point 0, we n=1 an n define three abscissas σa , σu and σc , respectively, the abscissas of absolute, uniform and simple convergence in a vertical half-plane, with obvious definitions. Cahen and later Bohr proved the following formulas, which are the analogues of the Hadamard formula for the radius of convergence of a power series, namely defining A N , U N , A∗N (|A N | ≤ U N ≤ A∗N ) by AN =

N 

an ,

n=1

A∗N =

N 

|an |,

(4.2.1)

n=1

N N         an n −it  =  an n −it  , U N = sup  t∈R

n=1

n=1

∞

they proved the following. Theorem 4.2.1 The convergence abscissas σc ≤ σu ≤ σa of a Dirichlet series  ∞ −s diverging at 0 are given by the formulas n=1 an n log |A N | log U N , σu = lim sup , log N N →∞ log N log A∗N σa = lim sup . N →∞ log N

σc = lim sup N →∞

(4.2.2)

Moreover, we have σa − σc ≤ 1 and the constant 1 is optimal. Proof The first and third formulas are well-known and due to Cahen. Their proof can be found, for example, in [3]. We provide some details for the second formula, due to Bohr: let us put a = lim sup log U N / log N ≥ 0. N →∞

N an n −it and A0 (t) = 0, so that |A N (t)| ≤ Cε N a+ε . Take Let ε > 0 and A N (t) = n=1 s = σ + it with σ ≥ a + 2ε. We can write by Abel’s transformation: N  n=1

=

N −1  n=1

an n

−s

N  = [An (t) − An−1 (t)]n −σ n=1

An (t)[n −σ − (n + 1)−σ ] + A N (t)N −σ .

98

4 General Properties of Dirichlet Series

The last term tends uniformly to 0 on Ca+2ε since it is dominated by Cε N a+ε−a−2ε = Cε N −ε .  −σ − (n + 1)−σ ] is normally convergent on Ca+2ε since And the series ∞ n=1 An (t)[n its general term is dominated by

Cε n a+ε σn −σ−1 = Cε n a+ε−1 σn −σ ≤ Cε (a + 2ε)n −1−ε ≤ a + 2ε. Indeed, the function σ → σn −σ = σe−σ log n decreases on the interval [1/ log n, ∞[ because σ ≥ a + 2ε. This proves that σu ≤ a + 2ε and finally that σu ≤ a. The converse can be seen by using the famous Kronecker lemma (not to be confused with the Kronecker theorem), which we recall as follows.

for

1 log n

Lemma 4.2.2 Let (vn ) be a sequence of vectors of the normed space X and (λn ) be a non-decreasing sequence of positive numbers tending to +∞. We assume that the  v /λ converges in X . Then, we have series ∞ n n n=1 N  1    vn  = 0.  N →∞ λ N n=1

lim

The proof just consists of Abel’s transformation (once more), but this is the natural context here. If we apply this lemma to X , the space of bounded functions on R with the sup-norm, to vn ∈ X defined by vn (t) = an n −it and λn = n σu +ε , remembering that σu ≥ 0 by our hypothesis, we get from the lemma that U N /N σu +ε → 0. This implies that a = lim sup N →∞ log U N / log N ≤ σu + ε and finally a ≤ σu , ending the proof of (4.2.2). The last assertion comes from the following: Let ε > 0. Then, |an | ≤ Cε n σc +ε/2 , so that |an |n −σc −1−ε ≤ Cε n −1−ε/2 , 1 + ε and finally σa ≤ σc + 1. The optimality is seen with showing that σa ≤ σc + n−1 −s n for which σc = 0 and σa = 1. the alternate zeta series ∞ n=1 (−1) 

4.2.2 The Perron–Landau Formula  −s Let f (s) = ∞ be a Dirichlet series such that σc ( f ) < ∞. As in the case n=1 an n of Fourier series, it is natural to try to recover an , or its summatory function A(x) =  n≤x an , from the behaviour of f on some line. If the series converges uniformly on Re s = ρ, we can consider it as an almost periodic function on that line, and recover

4.2 Convergence Abscissas

99

an from the Fourier–Bohr formulas of Chap. 1. Now, the Cauchy formula allows us to do better. We first have the elementary Perron–Landau formulas, in which a > 0, the parameter T → +∞ and the O’s are absolute ([4], p. 342 or [5], p. 135):  a+i T s y 1 ds = O(y a /T | log y|) 0 < y < 1 =⇒ 2iπ a−i T s  a+i T s y 1 ds = 1 + O(y a /T log y). y > 1 =⇒ 2iπ a−i T s

(4.2.3) (4.2.4)

Using those relations, we have the “first effective Perron–Landau formula” as follows.  −s Theorem 4.2.3 (Perron–Landau formula) Let f (s) = ∞ be a Dirichlet n=1 an n series. Let ρ > max(0, σa ), T ≥ 1 and let x ≥ 1, not an integer. Then: A(x) =

1 2iπ



ρ+i T ρ−i T

f (s)

 xρ  |an | xs ds + O s T n≥1 n ρ | log(x/n)|

(4.2.5)

where the O is absolute. Proof Using (4.2.3) and (4.2.4) with y = x/n, we have, by absolute convergence on the line Re s = ρ:  ρ+i T  ρ+i T ∞  1 1 (x/n)s xs f (s) ds = an ds 2iπ ρ−i T s 2iπ ρ−i T s n=1    1 an 1 −1 ρ ) + O((x/n)ρ ) an 1 + O(T (x/n) = log(x/n) T | log(x/n)| n≤x n>x   xρ  |an | = A(x) + O , T n≥1 n ρ | log(x/n)| which ends the proof of Theorem 4.2.3.



The formula (4.2.5) will be very useful in the study of the space H∞ of bounded Dirichlet series in Chap. 6, and in Chap. 7 as well. It can be found in [5], p. 135, but we gave its full proof to emphasize that everything is contained in (4.2.3) and (4.2.4).

4.2.3 The Holomorphy Abscissa A fourth abscissa, the holomorphy abscissa σh , defined as the farthest left abscissa of the vertical half-plane in which the sum of the series (4.1.1) has a holomorphic extension, plays a key role for Dirichlet series. Indeed, while a power series always

100

4 General Properties of Dirichlet Series

has a singular point on its circle of convergence, a Dirichlet series may very well have no singular point on its line of convergence Re s = σc , as is shown by the alternate zeta function (a special case of Theorem 4.2.5): A(s) =

∞  (−1)n−1 n −s = (1 − 21−s )ζ(s).

(4.2.6)

n=1

For that series, we clearly have σc = 0, whereas if you know the properties of zeta, you see that the zero of 1 − 21−s at 1 kills the unique pole of ζ at this point. But this requires the analytic extension of zeta. Here is a quite striking and more elementary example. We denote by (wn )n≥0 the Thue–Morse sequence inductively defined by w0 = 1; w2n = wn ; w2n+1 = −wn . The associated generating function is ∞ 

wn q n =

∞ 

n=0

Let W (s) =

∞ n=0

j

(1 − q 2 ), |q| < 1.

(4.2.7)

j=0

wn (n + 1)−s , Re s > 1. Then, we have the following theorem.

Theorem 4.2.4 The simple convergence abscissa of W is σc = 0. The holomorphy abscissa of W is σh = −∞. Proof Let Wn = w0 + · · · + wn . We see that W2n+1 = 0 and |W2n | = 1. Therefore, the formula (4.2.2), applicable since the series diverges at 0, gives σc = 0. For the analytic extension, we make use of the obvious integral formula −s

(n + 1)

1 = (s)

Using (4.2.8), (4.2.7), and setting W (s) =

1 (s)

 0

∞

t s−1 e−t



n=0

t s−1 e−(n+1)t dt, Re s > 0.

(4.2.8)

0



∞ 

∞

(t) =

∞

j=0 (1

 wn e−nt dt =

− e−2 t ), we obtain for Re s > 1:

1 (s)

j



∞

t s−1 e−t (t)dt. (4.2.9)

0

Now,  the RHS of (4.2.9) is an entire function, since the zero of infinite multiplicity of compensates the singularity of t s−1 at the origin, for any s ∈ C. This ends the proof. 

4.2 Convergence Abscissas

101

4.2.4 A Class of Examples Here are some other examples. We will consider the convergence and possible analytic continuation inside a class of special Dirichlet series, involving diophantine approximations to some irrational real number θ. An element of this class is the famous Hecke series (where x denotes the distance of x to the nearest integer): ζθ (s) =



nθ n −s , Re s > 1.

n≥1

Hecke showed that this series has a meromorphic continuation to the whole plane when θ is a quadratic irrational number. A few years later, Hardy and Littlewood discovered a class of well approximated numbers for which the series admits Re s = 1 as a natural boundary. We now focus on some extensions of such results. In [6], the Dirichlet series of the following form are considered: ζg,θ (s) =



g(nθ)n −s

n≥1

where θ ∈ R, and g is some 1-periodic, piecewise continuous function. We shall prove the three following results, the last two of which are issued from [6]. Theorem 4.2.5 2, (an ) a non-zero q-periodic sequence with q Let q be an integer ≥ −s mean value j=1 a j = 0 and A(s) = ∞ n=1 an n . Then, we have σc = 0 and σh = −∞. Proof of Theorem 4.2.5 : The  relation σc−s= 0 is clear. For σh , we might use the Hurwitz zeta function ζa (s) = ∞ n=0 (n + a) , 0 < a ≤ 1, which has the same properwritten under this form ζa (s) = 1/(s − 1) + E a (s), ties as ζ = ζ1 in that it can be  q with E a an entire function. As r =1 ar = 0, we get the easily computed expression (since akq+r = ar ): A(s) = q −s

q  r =1

ar ζ qr (s) = q −s

q 

ar E qr (s),

r =1

and the extension follows. But we rather proceed, as we shall do in the next theorem, by using a polylogarithmic function instead of Hurwitz ones. Let ω = e2iπ/q , then let G = {ω j , 0 ≤ j ≤ q − 1} be the group of qth roots of unity and  = Z/qZ its dual. We define f : G → C by f (ω k ) = ak . Its Fourier and inverse Fourier transforms are q−1 q−1  1   ak ω jk , an = f ( j)ω n j , n ≥ 1. f ( j) = q k=0 j=0

102

4 General Properties of Dirichlet Series

Thanks to this representation of an , and the fact that  f (0) = 0, we obtain for Re s > 1 the following expression: A(s) =

q−1 

 f ( j)L(ω j , s)

j=1

where L(z, s) =

 zn , Re s > 0, |z| ≤ 1, z = 1, ns n≥1

is the polylogarithmic function to be considered in the second theorem, and which is entire (for fixed z with |z| ≤ 1 and z = 1), as we will prove. Now, the result follows since ω j = 1 for 1 ≤ j ≤ q − 1. Note that this situation typically occurs when an = χ(n), where χ is a non-principal character mod q, for some integer q ≥ 2. The corresponding Dirichlet series is the L-function L(s, χ) =

∞ 

χ(n)n −s

n=1



which extends to an entire function.

∞ Theorem 4.2.6  1 Let g be any non-zero 1-periodic function in C (R) with meanvalue  g (0) = 0 g(t)dt = 0 and θ be a diophantine number. Then the abscissa of convergence σc of ζg,θ is 0 and ζg,θ has an analytic continuation to the entire complex plane, i.e. σh = −∞.

Theorem 4.2.7 Let g be any 1-periodic function with bounded variation and zero mean, and θ be an r -diophantine number. Then we have σc ≥ 1 − 1/r . the formula (4.2.2) for the convergence abscissa of a Dirichlet series  We recall −s a n diverging at s = 0: n n≥1 σc = lim sup log | N →∞

N 

an |/ log N .

(4.2.10)

n=1

As a result, the description of the convergence domain rests on an estimate of the additive cocycle  S N := g(nθ). 1≤n≤N

(By abuse of language, g itself is referred to as a “cocycle”.) Let E be some space of functions on the topological space X , and T a continuous transformation on X . Recall that g ∈ E is a “coboundary” (in E) for the transformation T if there exists h ∈ E such that

4.2 Convergence Abscissas

103

g(x) = h(T x) − h(x).

(4.2.11)

Proof of Theorem 4.2.6 : By assumption, g ∈ C ∞ (R) has zero mean, its Fourier coefficients verify g(n) ˆ = O(|n|− p ) for any p ≥ 1 and qθ ≥ δq −r for some constants r ≥ 1 and δ > 0. We derive easily from those observations that g must be a “coboundary” for the rotation T : x → x + θ on T: indeed, putting cn =

g(n) ˆ , n ∈ Z∗ e2iπnθ − 1

clearly ˆ ≤ C|n|r |g(n)| ˆ |cn | ≤ |g(n)|/ nθ

so that (cn ) is the Fourier sequence of some 1-periodic function h ∈ C ∞ (R), satisfying h(x + θ) − h(x) = g(x). (4.2.12) In view of the previous comments, |S N | = |



g(nθ)| = |h[(N + 1)θ] − h(θ)| ≤ 2 h ∞

1≤n≤N

and the first assertion follows from (4.2.10). The second part of the theorem is a consequence of integral formulas valid for the polylogarithms L(z, s) =

 zn , Re s > 0, |z| ≤ 1, z = 1. ns n≥1

Recall the identity (4.2.8) n −s = L(z, s) =

1 (s)

∞

z (s)

0



u s−1 e−nu du, which implies

∞ 0

u s−1

e−u du, 1 − ze−u

providing an analytic continuation of s → L(z, s) to the half-plane Re s > 0 when z = 1. Then, in order to increase the half-plane of analyticity, we proceed this way:  ∞ z du du = u su s−1 u u e −z s(s) 0 e −z 0  ∞ eu z s u u du = (s + 1) 0 (e − z)2

z L(z, s) = (s)



∞

s−1

(thanks to integration by parts) which is analytic for Re s > −1, so we got an additional strip for analyticity. After r similar steps, we can write

104

4 General Properties of Dirichlet Series

L(z, s) =

z (s + r )



∞

u s+r −1

0

Pz (eu ) du − z)r +1

(eu

where Pz is a polynomial of degree r such that |Pz (eu )| ≤ Cer u uniformly in |z| = 1. For z = e2iπkθ , it follows from (4.2.14) that   r −(r +1)u/2   Pz (eu )  in a neighbourhood of 0 ≤ Ck e ϕ(u) :=  u (e − z)r +1  outside C e−u    s+r −1 Pz (eu )   ≤ u Re s+r −1 ϕ(u) u  (eu − z)r +1 

and

has to be integrable for Re s > −r . Moreover, if K denotes a compact subset of C−r , it holds: |L(e2iπkθ , s)| = O(k r ) uniformly in s ∈ K . The Weierstrass theorem terminates the proof of the analytic continuation of L to that half-plane. Finally, integrating by parts as many times as needed, we get an analytic continuation to the entire complex plane (provided z = 1) that we still denote by L. Now, if Re s > 0, we have     2iπknθ n −s g(nθ)n −s = g(k)e ˆ ζg,θ (s) = n≥1

=



k∈Z∗

g(k) ˆ



n≥1

e

2iπnkθ −s

k∈Z∗



n

=



2iπkθ g(k)L(e ˆ , s),

(4.2.13)

k∈Z∗

n≥1

because of the pretty good convergence of the Fourier series of g. For every k = 0, L(e2iπkθ , ·) has an analytic continuation to the whole complex plane; moreover, an easy computation leads to |1 − e2iπkθ t|2 ≥ 4t sin2 (πkθ) ≥ 16t kθ 2 , 0 < t ≤ 1 so that, θ being diophantine, we have √ |1 − e2iπkθ t| ≥ δk −r t for some r ≥ 1 and δ > 0. We obtain |L(e2iπkθ , s)| ≤

Ck r |(s)|



1 0

dt (log(1/t))Re s−1 √ t

(4.2.14)

4.2 Convergence Abscissas

105

and the uniform convergence of the series (4.2.13) of entire functions in restriction to any compact set of s ∈ {Re s > 0}. If K is a compact subset of C, K ⊂ {Re s > −r } for some positive integer r and, as above, we replace L by its analytic continuation to this half-plane. This ends the proof of Theorem 4.2.6.  Proof of Theorem 4.2.7 : In the case where g is only of bounded variation, g has no more reason to be a coboundary, whatever θ is: actually, even when g is real analytic, Herman showed that the measurable solution h of the cohomological equation (4.2.12) does not need to be bounded [7]. Thus, we have to handle with more general estimates on cocycles as those given by the classical Denjoy–Koksma inequality that we establish now [7]. The statement relies on the continued fraction expansion of θ (cf. Chap. 3). Proposition 4.2.8 (Denjoy–Koksma) Let g be a real 1-periodic function with N g(nθ) as above. bounded variation Var (g) and zero mean. Consider S N = n=1 Then, |Sq | ≤ Var(g) for any denominator q of θ. As a consequence, 1. If θ ∈ Bad, S N = O(log N ). 2. If θ is r -diophantine for some r > 1, |S N | ≤ C N 1−1/r (log N ). Note that the inequality is of no use when r → ∞ (θ Liouville). Proof Let p/q be a convergent of θ satisfying, without loss of generality, p 1 p ≤ θ < + 2, q q q and consider the q arcs I j = [ j−1 , qj [, j = 1, . . . , q. We first remark that each I j q contains one unique point of the finite orbit θ, 2θ, . . . , qθ (mod 1) denoted by y j := n j θ. Indeed, since ( p, q) = 1, we can choose n ∈ {0, 1, . . . , q − 1} and k ∈ Z such that np + kq = j − 1. Since 0 ≤ nθ − n qp < qn2 < q1 , it follows that 0 ≤ nθ + k −

j −1 j < q q

 whence the claim with n j := n. Now put m j := q I j g(x)d x for j = 1, . . . , q; 1 q q reminding j=1 m j = q 0 g(x)d x = 0, we get Sq = j=1 (g(y j ) − m j ) with  g(y j ) − m j = q

(g(y j ) − g(x))d x Ij

so that

q    |Sq | = q  j=1

0

1/q



g(y j ) − g(x +

j − 1   ) dx q

106

4 General Properties of Dirichlet Series

 ≤q

1/q

0

 1/q q   j − 1   )d x ≤ q Var(g)d x = Var(g), g(y j ) − g(x + q 0 j=1

the expected upper bound. The same holds if qp − q12 < θ ≤ qp . We write as usual θ = [0; a1 , a2 , . . .] and pn /qn = [0; a1 , a2 , . . . , an ]. Fix now N and qt such that qt ≤ N < qt+1 . We can expand N as N = b0 + b1 q1 + · · · + bt qt where b j < q j+1 /q j . Thus, |S N | ≤ (b0 + b1 + · · · + bt )Var(g).

(4.2.15)

If θ ∈ Bad, the sequence (q j ) is at most lacunary and b j ≤ K for some positive constant K so that |S N | ≤ K (t + 1)Var(g) ≤ C log N Var(g). If θ is r -diophantine, there exists a positive constant K such that q j+1 /q rj ≤ K , or 1/q j ≤ (K /q j+1 )1/r , holds for every j, from which (with unexplicit constants) bj
0 such that A(x) ≤ C x(log qt +

qt+1 ) ≤ C x(log qt + at+1 ) qt

where qt is such that qt ≤ x < qt+1 . Proof In the Euclidean division of a by b, we choose the multiple of b nearest to a. (a) We establish first a minoration of nθ . Let p/q be a convergent of θ, in particular |θ − p/q| ≤ 1/q 2 with ( p, q) = 1, and fix n, 1 ≤ n ≤ q. Then θ= and nθ =

ε p + 2 , |ε| ≤ 1 q q

nε np nε v + 2 = k ± + 2 =: k + r q q q q

if np = kq ± v with v ∈ [0, q/2). Assume 2 ≤ v ≤ q/2 and r = v/q + nε/q 2 ; then it is easily seen (n ≤ q) that v−1 v−1 ≤r ≤1− q q so that

nθ = min(nθ − k, k + 1 − nθ) = min(r, 1 − r ) ≥ (v − 1)/q. (The same holds if v ≤ −2). (b) Let now t be such that qt ≤ x < qt+1 ; recall that (3.1.1): 1 1 ≤ qt θ ≤ ; 2qt+1 qt+1 since n ≤ qt+1 , the optimality property of the convergents implies

nθ ≥ qt θ >

1 for 1 ≤ n ≤ x. 2qt+1

108

4 General Properties of Dirichlet Series

We first restrict mqt < n ≤ (m + 1)qt ; observe that (npt ) runs over a complete residues set mod qt so that v takes all values between −qt /2 and qt /2. If v = 0, ±1 we have nθ ≥ 1/2qt+1 as already quoted. If |v| ≥ 2, we may apply the inequality obtained in (a) and get nθ ≥ v − 1/qt (for positive v). Finally, 

 1 qt ≤2  qt log qt

nθ

v −1 2≤v≤q /2 |v|≥2 t

and

 n≤x mqt 0, one can construct gradually a sequence an such that log an+1 / log qn approaches λ = σ − 1. If λ > 0 and given a0 , an+1 = λ. a1 , . . . , an so that qn is determined, we choose next an+1 = [qnλ ]; thus lim log log qn n If we choose an+1 = [qn ], then σc = ∞. (2) We have an evident minoration of qn for any θ: by induction qn ≥ Cτ n where n−1 τ is the golden number (also qn ≥ 2 2 ). If θ ∈ B, log an+1 / log qn  1/n → 0. If θ is any algebraic number, we can deduce from Roth’s theorem a control on the partial quotients: indeed, fix 1 > ε > 0; there exists a constant δε such that the inequality |x − qp | ≥ δε /q 2+ε holds for every rational p/q; besides, from (3.1.1), |x − qpnn | ≤ 1/qn qn+1 so that, for the convergents, δε qn1+ε

≤

1 qn+1

and an+1 = O(qnε ). This gives (2), ε being arbitrary small. (3) Now observe that an (x) = O(n 2 ) for almost every x. Indeed, the Gauss measure μ being T -invariant, we have μ({x; an (x) > n 2 }) = μ({x; a1 (T n−1 x) > n 2 }) = μ({x; a1 (x) > n 2 }). 2 2 But a1 (x) > n 2 means [1/x] > n or x < 1/n 2. As μ is equivalent to the Lebesgue measure, it follows that n μ({x; an (x) > n }) < ∞, and by Borel–Cantelli’s lemma, for almost all x, an (x) ≤ n 2 ultimately. Thus (3) follows from the sure lower bound of qn . We could also invoke the ergodic theorem. Taking f = log a1 , this ergodic theorem implies that a.e.

log a1 + · · · + log an → n since f ∈ L 1 (X, μ). It follows that

log an n

 f (x) dμ(x) < ∞ X

→ 0 a.e.



110

4 General Properties of Dirichlet Series

4.2.6 Automatic Dirichlet Series We enlarge the notion of holomorphy abscissa by defining σm , the meromorphy abscissa of a Dirichlet series. We limit ourselves to a class of coefficients arising from symbolic dynamics. Recall that the open set Ω is a meromorphic domain for the function f if f cannot be extended meromorphically to a strictly larger open set than Ω. Carlson [9] has considered the case when Ω = D(0, 1) and f is a power series with coefficients taking values in a finite subset of C; he then proved they admit a finite set of singularities on the unit circle if and only if the sequence of coefficients is ultimately periodic. A few years later, a theorem due to Polya–Carlson asserted that a power series with coefficients ∈ Z and radius of convergence R = 1 either admits D as a holomorphy domain or can be extended to a rational function of the form P(z)/(1 − z m )n where P(x) ∈ Z[x]. Such a result does not hold with Dirichlet series as it can readily be seen with ±1 coefficients.  −s with We are interested now, more generally, in the Dirichlet series ∞ n=1 an n coefficients in a finite subset of C; of course they all converge for s > 1, but various situations may appear. To our knowledge, the class of automatic Dirichlet series appeared first in [10], apart of specific examples. Roughly speaking, a sequence with values in a finite set is d-automatic if it can be computed, term by term, by using a finite-state machine and the base d ≥ 2 expansion of numbers. The following concise definition avoids the tedious introduction of a finite-state automaton. Definition 4.2 A sequence (u n )n≥0 with values in the finite set A is d-automatic if its d-kernel Nd (u) is finite where the d-kernel is defined by Nd (u) = {n → u d k n+a , k ≥ 0, 0 ≤ a ≤ d k − 1}. Examples 1 This definition applies to the Thue–Morse automatic sequence u (on {±1}): in this case, d = 2 and N2 (u) = {±u}; actually, u is a 2-multiplicative sequence satisfying u 0 = 1, u 2n = u n , u 2n+1 = −u n , so that u 2k n+a = u n u a for all k ≥ 0, 0 ≤ a < 2k . 2. The Rudin–Shapiro sequence [11] is a sequence with values in {±1} defined by u n = (−1) f (n) where f (n) counts the number of 11 in the decomposition of n to base 2; u satisfies u 0 = 1, u 2n = u n , u 2n+1 = (−1)n u n =: vn . Observe that u 2k n+a = u n u a if 0 ≤ a < 2k−1 , whence  u n u b if 0 ≤ b < 22k−1 u 4k n+b = vn u b if 22k−1 ≤ b < 22k . It follows that N4 (u) = {u, −u, v, −v} and u is a 4-automatic sequence. Let d ≥ 2 and (u n )n≥0 a d-automatic sequence (thus taking finitely many values in, say, A); we put

4.2 Convergence Abscissas

111

f (s) =

 n≥0

un , s > 1. (n + 1)s

Before stating a continuation result for f , it is convenient to deal with the sequences of the kernel globally and note the following. Proposition 4.2.9 There exists an integer q ≥ 1 and a sequence of column vectors (Un )n≥0 with values in Aq such that: (i) the first component of the vector (Un )n≥0 is the sequence (u n )n≥0 ; (ii) for every 0 ≤ j ≤ d − 1 and n ≥ 0, Udn+ j = A j Un where the A j are q × q row-stochastic matrices with 0–1 entries.  Un Now consider for s > 1 the Dirichlet vector F(s) = ∞ n=0 (n+1)s · Thanks to Proposition 4.2.9, we decompose F(s) =

∞ d−1   j=0 n=0

=

∞ d−2   j=0 n=0

d−1 ∞

 Udn+ j A j Un = (dn + j + 1)s (dn + j + 1)s j=0 n=0 ∞

 Ad−1 Un A j Un + , s (dn + j + 1) d s (n + 1)s n=0

which leads to (I − d −s Ad−1 )F(s) =

d−2  j=0

Aj

∞  n=0

Un · (dn + j + 1)s

(4.2.18)

(here I is the q × q unit matrix). Let us now set M(s) = I − d −s (A0 + A1 + · · · + Ad−1 ).

(4.2.19)

Identity (4.2.18) provides a meromorphic continuation of F to the half-plane s > 0 by observing that M(s)F(s) =

d−2 

Aj

j=0

or else M(s)F(s) =

∞ 

 Un

n=0

d−2  j=0

Aj

   Un 1 − d −s Aj , s (dn + j + 1) (n + 1)s n=0 j=0

∞  n=0

d−2

 Un

∞

 1 1 − (dn + j + 1)s (dn + d)s

112

4 General Properties of Dirichlet Series

with poles at s annihilating det M(s). In order to proceed, we expand by the negative binomial formula: ∞  n=0

  Un Un (d − j − 1)k  k = C (dn + j + 1)s d s (n + 1)s k=0 s+k−1 d k (n + 1)k n=0

=

∞

∞ 

∞

∞

k Cs+k−1 (d − j − 1)k

k=0

It ensues that M(s)F(s) =

d−2  j=0

 F(s + k) F(s + k) =: αk, j . s+k d d s+k k=0

Aj

∞  k=1

αk, j

F(s + k) · d s+k

(4.2.20)

We deduce from (4.2.20) the following theorem. Theorem 4.2.10 Let d ≥ 2, (u n )n≥0 be a d-automatic sequence, as well as F(s) =  ∞ −s a vector of Dirichlet series. Then F has a meromorphic continn=0 Un (n + 1) uation to the whole complex plane with poles (if any) located on a finite number of left semi-lattices. Indeed, the right-hand term in (4.2.20) is absolutely convergent and holomorphic in s > 0, so that F can be meromorphically extended to this half-plane with (possible) poles located at the singular points of M(s), namely 

s ∈ C : d s is an eigenvalue of A :=

d−1 

 Aj .

j=0

Then we iterate the process to get the theorem.  Back to examples. 1. In the first Thue–Morse example, we have d = 2, Un = (u n , −u n ), A0 = I and A1 = −I so that M(s) = I , and there is no singularity at all. We recover σh = −∞. sequence admits a zero mean, more precisely, A N := √  2. The Rudin–Shapiro n 0) was given in [13], with a method which reproves Theorem 4.3.2 in a simple way. Theorem 4.3.3 The ring Rc of convergent Dirichlet series is a factorial ring. Two important elements of R are e defined by e(n) = 1 for all n ∈ N and its inverse for ∗, and the Möbius function μ defined by μ(1) = 1, μ(n) = (−1)k if n = p1 . . . pk ,

(4.3.2)

μ(n) = 0 otherwise. With this definition, in which p1 , . . . , pk denote distinct primes, it is easily checked that μ = e−1 , namely,  d|n

μ(d) = 1 if n = 1 and

 d|n

μ(d) = 0 otherwise.

114

4 General Properties of Dirichlet Series

The quite artificial-looking definition of μ is of course motivated by the zeta function and its Euler product: ζ(s) =

∞ 

e(n)n −s =

n=1



(1 − p −s )−1

p

so that, using Eq. (4.3.1) ∞

  1 = μ(n)n −s = (1 − p −s ) ζ(s) p n=1 and expanding formally the RHS, we fall on the definition (4.3.2) of the Möbius function. It is easily seen that Theorem 4.3.1 extends to the case where A is absolutely convergent and B convergent at s0 , and the series C(s0 ) may diverge if A and B are just convergent at s0 (see exercise). But we might expect C to converge at s0 + ε for each ε > 0. This is not the case, and the following theorem may appear as a surprise.   −s −s Theorem 4.3.4 If A(s) = ∞ and B(s) = ∞ converge at s = 0, n=1 an n n=1 bn n 1 then the product series C converges at 2 + iτ = s, where τ is real. And the result is optimal as concerns Re s. Proof We first prove the optimality. Let E be the Banach space of convergent series a = (an ), equipped with the norm

a = sup |An | where An = a1 + · · · + an . n≥1

And let (L n ) be the sequence of continuous bilinear forms on E defined by L n (a, b) =

c1 + · · · + cn ϕn

(4.3.3)

where c = a ∗ b and (ϕn ) is a fixed increasing sequence of positive numbers. We shall prove that   cn c ∈ E, If a, b ∈ E =⇒ = ϕ ϕn √ then ϕn ≥ δ n for some δ > 0.

(4.3.4)

Indeed, the assumption and the Kronecker lemma on series imply that L n (a, b) tends to 0 for all a, b ∈ E. The Banach–Steinhaus theorem for continuous bilinear forms on E (which can be viewed as continuous linear maps: E → E ∗ , where E ∗ is the dual of E) now provides a constant M > 0 such that

4.3 Products of Dirichlet Series

115

|L n (a, b)| ≤ M a b for all a, b ∈ E. n ai B[n/i] where Bm = b1 + · · · + bm and [ ] denotes the Since c1 + · · · + cn = i=1 integer part, this reads as well (fixing n, and setting λi = [n/i]): n     ai Bλi  = |c1 + · · · + cn | ≤ M a b ϕn for all a, b ∈ E. 

(4.3.5)

i=1

Taking for n a perfect square, observe that 1≤i < j ≤

√ n < k =⇒ λi > λ j > λk .

It suffices to see that n n n n n n − ≥ − = ≥ 2 ≥1 i j j −1 j j ( j − 1) j and moreover that λk < b so as to have

√ √ n ≤ λ j , since n is an integer. This allows us to choose

Bλi = sign ai if i ≤

√

n,

Bλi = 0 if i >

√

n, b = 1.

Testing (4.3.5) on that b, we get √

n 

|ai | ≤ Mϕn a .

i=1

√ √ Finally, take√ai = (−1)i if i ≤ n, ai = 0 otherwise to get n ≤ Mϕn for n a perfect square and n − 1 ≤ Mϕn for any n since ϕn is increasing. This proves (4.3.4), and therefore the optimality in Theorem 4.3.4. For the convergence at 1/2, or even at a point 1/2 + iτ = s0 with τ ∈ R, we need some notation: ∞  ai A0 = , s0 i i=1

∞  bj B0 = , α = sup |ai |, β = sup |b j |, j s0 j=1

εb (x) = sup | y≥x

 bj  ai |, ε (x) = sup | |. a j s0 i s0 y≥x x< j≤y x 0. Since our series converges uniformly in the whole half-plane Cσ , we can find an integer N such that

n 

a j j −s ∞ ≤ sup | f (s)| + ε ≤ f ∞ + ε s∈Cσ

j=1

for any n ≥ N . From the first part of the proof, it follows that 

|a p | p −σ ≤ Sn ∞ ≤ f ∞ + ε.

p≤n

Letting ε tend to 0 and then n to ∞ and σ to 0, we get the result.



4.4 Bohr’s Abscissa via Kronecker’s Theorem

119

Bohr derived a second inequality from his method, under the form of the following theorem, even if we shall see later a more natural and direct method.  −s Theorem 4.4.2 For any Dirichlet series ∞ n=1 an n , we have the inequality 1 σa ≤ σu + . 2

(4.4.3)

 −σ it Proof Let ε > 0 and σ = σu + ε. By definition, the series ∞ n converges n=1 an n

≤ C, where S N (t) = uniformly on R, so there exists a constant C such that

S N ∞ n it a n . Now, (4.4.2) implies n n=1

S N 2 ≤ S N ∞ = S N ∞ ≤ C. N 1 (|an |2 n −2σ ) 2 ≤ C, whence By relation in Tr , this reads as well n=1  ∞Parseval’s 2 −2σ −τ < ∞. Now, if τ = σ + 21 + ε, the series ∞ is convergent n=1 |an | n n=1 |an |n by the Cauchy–Schwarz inequality, as the pointwise product of the two square1 summable series |an |n −σ and n − 2 −ε . This gives σa ≤ τ = σu + 21 + 2ε and (4.4.3) by letting ε tend to 0.  ∞ If a Dirichlet series n=1 an n −s converges uniformly in C0 , for example, it follows from the preceding that (1) It converges absolutely in C0 if an = 0 for n = p. (2) It always converges absolutely in Cσ if σ > 21 . Therefore, Bohr was naturally led to ask the two following questions : (1) Is it always true that we have absolute convergence in C0 ? (2) If not, what is the maximum T of the difference σa − σu ? The answer to the first question is No (see Example 4). But this does not even imply that T > 0! The answer to the second question is that T = 1/2. This is a highly non-trivial result [17] even if it is old. The proof will be detailed in Chap. 6.

4.4.3 A Wiener Lemma for Dirichlet Series Every Fourier analyst  knows the celebrated “lemma” of Wiener, saying that if f (t) =  int with n∈Z |an | < ∞ and f (t) = 0 for all t ∈ R, then n∈Z an e   1 = bn eint with |bn | < ∞. f (t) n∈Z n∈Z Here is the analogue for Dirichlet series, due to Hewitt and Williamson [18], and whose proof is slightly more involved.

120

4 General Properties of Dirichlet Series

  −s Theorem 4.4.3 Let f (s) = ∞ with ∞ n=1 an n n=1 |an | < ∞ and suppose that | f (s)| ≥ δ > 0 for s ∈ C0 . Then, one can write, for s ∈ C0 : ∞ ∞   1 = bn n −s with |bn | < ∞. f (s) n=1 n=1

Proof The shortest proof of Wiener’s lemma consists in noticing that the assumption f (t) = 0 reads δt ( f ) = 0 for all t ∈ R/2πZ =: T and that the set of evaluations δt , t ∈ T is exactly the spectrum of the Wiener algebra W of absolutely convergent Fourier series. Therefore, the Gelfand transform of f does not vanish, and Gelfand’s theory shows that f is invertible in W , which was to be proved. Here, the situation is more complicated: it is easy (and was already mentioned in Chap. 1) to see that the spectrum M of the Banach algebra A of absolutely  ∞ −s a n such that

f

:= convergent Dirichlet series f (s) = ∞ n=1 n n=1 |an | < ∞, equipped with the Dirichlet multiplication, can be identified with the set S of completely multiplicative non-zero functions χ : N∗ → C such that |χ(n)| ≤ 1 ∀n ∈ N∗ by the formula ∞   an χ(n). f (χ) = n=1

Examples of functions χ are the functions χs , s ∈ C0 with χs (n) = n −s . But, contrary to the Wiener case, such functions are far from exhausting the spectrum M, or even from being dense in M, and rather appear as a boundary of M. For the proof of Theorem 4.4.3, we will need some notations (in which r is a fixed positive integer) and a lemma. Put U = Dr = {z = (z 1 , . . . , zr ); max1≤ j≤r |z j | < 1}. = {z = (z 1 , . . . , zr ); |z j | < p −σ Uσ j , 1 ≤ j ≤ r }, 0 ≤ σ ≤ ∞. ∂0 U σ = {z = (z 1 , . . . , zr ); |z j | = p −σ j , 1 ≤ j ≤ r }, 0 ≤ σ ≤ ∞ = distingusihed boundary of Uσ . With those notations, we have the following “distinguished minimum principle”. Lemma 4.4.4 Let Q(z) be a polynomial in r variables, and ρ > 0. Assume that |Q(z)| > ρ for all σ ≥ 0 and for all z ∈ ∂0 Uσ . Then: |Q(z)| > ρ for all z ∈ U. Proof of the Lemma: Observe that Uσ increases from {0} to U as σ decreases from ∞ to 0. Let m(σ) = inf z∈Uσ |Q(z)|. Suppose that m(σ) = ρ for some σ > 0. Then,

4.5 A Theorem of Bohr and Jessen on Zeta

121

1/Q is holomorphic on Uσ and the distinguished maximum principle applied to that function shows that inf |Q(z)| = inf |Q(z)| = ρ z∈Uσ

z∈∂0 Uσ

contradicting our assumption. Therefore, the continuous function m does not take the value ρ and since m(∞) = |Q(0)| > ρ, we have m(0) > ρ, proving the lemma.   Let now χ ∈ S, 0 < ε < δ and N such that ∞ |a | ≤ ε/2. Let r = π(N ), N +1 n N −s P(s) = n=1 an n , Q(z) = P(z) and ρ = δ − ε. We have |P(s)| ≥ | f (s)| − ε/2 ≥ δ − ε/2 for z ∈ C0 and so, by Kronecker’s theorem: |Q(z)| ≥ δ −

ε > ρ for z ∈ ∂0 Uσ , σ ≥ 0. 2

By the lemma, we have |Q(z)| ≥ ρ for all z ∈ U . In particular, N     an χ(n) = |Q(χ( p1 ), . . . , χ( pr ))| ≥ ρ  n=1

and

∞ N ∞          | f (x)| =  an χ(n) ≥  an χ(n) − |an | ≥ δ − 2ε. n=1

n=1

n+1

By letting ε tend to 0, we get |  f (χ)| ≥ δ. In particular,  f (χ) = 0 and now Gelfand’s theory proves that f is invertible in A, which was to be proved. 

4.5 A Theorem of Bohr and Jessen on Zeta We finish this chapter on the Dirichlet series with the proof by Landau [19] of a simplified version of a result of Bohr and Jessen. Other density results will appear in Chap. 7. Theorem 4.5.1 The image of the open half-plane C1 under the Riemann zeta function is the whole complex plane deprived of the origin: ζ(C1 ) = C\{0}. The proof relies on a Landau–Schnee theorem which we admit here [19].  −s with convergence, and moreover Theorem 4.5.2 Assume that f (s) = ∞ n=1 an n non-vanishing of f , for 1 < σ ≤ ∞, so that in particular a1 = 0 and |an | ≤ Cε n 1+ε ∀ε > 0.

122

4 General Properties of Dirichlet Series

Then we have ∞

σ > 1 =⇒

 1 bn n −s with |bn | ≤ Cε n 1+ε ∀ε > 0. = f (s) n=1

(4.5.1)

The second fact needed is the extension or Bohr’s device to other functions than Dirichlet polynomials: Let D0 be the set of all Dirichlet series f (s) =

∞ 

an n −s with |an | ≤ Cε n 1+ε ∀ε > 0

n=1

and, for each integer N ≥ 1, let  N = {z = (z 1 , . . . , z N ); |z j |
0. Then, the Riemann ζ function takes any complex, non-zero, value, in the small strip S = {s ∈ C; 1 < Re s < 1 + δ}. But we restricted ourselves to the statement of Landau, because the proof is a direct and striking application of the Bohr operators  N .

124

4 General Properties of Dirichlet Series

4.6 Exercises  1. A sequence (μn )n≥1 is called a multiplier if the series n≥1 an μn converges as  soon as n≥1 an converges. Show that the multipliers are exactly the sequences with bounded variation. (Hint: you can use the Banach–Steinhaus theorem.) 2. Let A, B be two Dirichlet series and C = AB their product. We assume that A converges at −1  and that B converges at 0. cn converges. (i) Show that n≥2 log n (ii) Using the Banach–Steinhaus theorem as in the proof of Theorem 4.3.4, show that the factor log n is optimal.  in α −s n , where 0 < α < 1. 3. We consider the Dirichlet series ∞ n=1 e (i) Using the Euler–Maclaurin summation formula, show that N  k=1

α

eik ∼

n 1−α in α e as n → ∞. iα

(ii)  Show thatthe abscissas of convergence verify σa = σu = 1 and σc = 1 − α. ∞ 4. Let ∞ n=1 an , n=1 bn be two convergent  series with sums A and B, and c = a ∗ b their Dirichlet product. Suppose that ∞ n=1 cn converges with sum C. Show that C = AB. (Hint: use the absolute convergence of the generating series of an , bn , cn for Re s > 1 and the principle of analytic continuation.) 5. Give an example of a convergent series ∞ n=1 an such that its Dirichlet square is a divergent series. (Hint: take an = εn χ(n) where εn decreases to 0 and χ is a non-trivial character.) 6. Let χ be a non-trivial character modulo some integer q ≥ 2 (the definition of such characters being taken for granted here), and L(s, χ) =

∞  χ(n) n=1

ns

.

Compute the abscissas σa , σu , σc , σh of this Dirichlet series. 7. μ be the Möbius function (whose basic properties are here taken for granted), Let ∞ −s n=1 μ(n)n (= 1/ζ(s)) the associated Dirichlet series, and σa , σu , σc the corresponding abscissas. (i) Show that σa = 1. 1 ≤ σc ≤ 1, and that the series diverges at s = 1/2. (ii) Show that ac verifies  2 1+it (iii) Show that the series ∞ converges uniformly on each compact n=1 μ(n)/n subset of R. Does this series converge uniformly on R? What is the value of σu ? −s with an = 0 for 8. Let P be the set of all Dirichlet polynomials P(s) = ∞ n=1 an n large n, equipped with the two norms

4.6 Exercises

125

P ∞ = sup |P(s)| and P W = s∈C0

∞ 

|an |.

n=1

We define inductively (in the Rudin–Shapiro manner) two sequences (Pn ), (Q n ) of Dirichlet polynomials by P0 (s) = Q 0 (s) = 1 and (( pn ) being the sequence of primes) −s Q n (s); Pn+1 (s) = Pn (s) + pn+1

−s Q n+1 (s) = Pn (s) − pn+1 Q n (s).

(1) Show that s ∈ C0 =⇒ |Pn+1 (s)|2 + |Q n+1 (s)|2 ≤ 2[|Pn (s)|2 + |Q n (s)|2 ]. n+1

(2) Show that Pn ∞ ≤ 2 2 and that Pn W = 2n . Prove that there are uniformly, non-absolutely convergent Dirichlet series in C0 . 9. Let θ be an irrational number. Extend the results of subsection 2.5 (i.e. compute σc and σa in terms of the convergents of θ) to the Dirichlet series ∞  n=1

n −s . sin nπθ

10. Consider the increasing sequence (λn ) of exponents defined by λ2n−1 = n and λ2n = n + e−n , 2

 n−1 −λn s e . as well as the Dirichlet series f (s) = ∞ n=1 (−1) (i) Observe that limn→∞ log n/λn = 0. In fact, λn ∼ n/2. (ii) Show that the convergence and the holomorphy abscissas of that series are different: σc = 0 and σh = −∞. (This example is due to Bohr.) Could this happen with λn = n? 11. Let us consider for s > 0  Fn−s , f (s) := n≥1

where Fn is the nth term of the Fibonacci sequence initiated by F1 = F2 = 1. It is well-known that Fn =

1 θ − θ¯

√ 1 1 √ 1 (θn − θ¯n ) = √ (θn − θ¯n ), θ = ( 5 + 1), θ¯ = (1 − 5). 2 2 5

126

4 General Properties of Dirichlet Series

Now, we denote by σm ( f ) the abscissa of the meromorphic extension of f and E( f ) the set of poles of f in Cσm ( f ) . We are interested in a description of those objects. As usual we write s = σ + it. √ 1. By approximating Fn by θn / 5, prove that θn Fn−s − ( √ )−s = |s|O(θ−n(σ+2) ). 5   θn −s −s √ 2. If now (s) = ∞ F , deduce from 1. that − ( ) n n=2 5 (s) → 1 and f (s) → 2 as |s| → ∞. 3. We shall prove that σm ( f ) = −∞ by

refining this approach. For s ∈ C and k ∈ N, we denote by ks the quantity s(s − 1) · · · (s − k + 1) ; k! observe that the formula (θ − θ¯n )−s = n

 k≥0

 (−1)

k(n+1)

 −s −n(s+2k) θ k

(4.6.1)

holds for s ∈ C. 4. By pumping (4.6.1) into the expression of f , show that   √  −s √  1 f k (s) := ( 5)s f (s) = ( 5)s s+2k + (−1)k+1 k θ k≥0 k≥0 for σ > 0 (Indication. Check that the exchange of summation in the double sum is valid). 5. Prove that we get this way a meromorphic extension of f to the open set Ω = C\E( f ) where E( f ) consists in the (simple) poles sk,n of the f k ’s to be explicited. (Inspired by [20].)

References 1. G.H. Hardy, M. Riesz, The General Theory of Dirichlet Series (Second Edition, Dover Phenix Editions, 2005) 2. H. Helson, Dirichlet Series (Regent Press, 2005) 3. E. Titchmarsh, The Theory of Functions (Clarendon Press, 1932) 4. E. Landau, Handbuch des Lehre von der Verteilung der Primzahlen, 2nd edn (Chelsea University Company, 1953)

References

127

5. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres (Institut Elie Cartan, 1995) 6. O. Knill, J. Lesieutre, Analytic continuation of Dirichlet series with almost periodic coefficients. Complex Anal. Oper. Theory 6, 237–255 (2012) 7. M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. No. 49, 5–233 (1979) 8. G.H. Hardy, J.E. Littlewood, Notes on the theory of series: a curious power series. Proc. Camb. Philos. Soc. 42(2), 85–90 (1946) 9. F. Carlson, Über Potenzreihen mit endlich vielen verschiedenen Koeffizienten. Math. Ann. 79, 237–245 (1918) 10. J.-P. Allouche, M. Mendès France, J. Peyrière , Automatic Dirichlet series. J. Number Theory 81, 359–373 (2000) 11. J.-P. Allouche, J. Shallit, Automatic sequences, Theory, applications, generalizations. Cambridge University Press, Cambridge, 2003. xvi+571 pp. ISBN: 0-521-82332-3 12. D. Cashwell, J. Everett, The ring of number-theoretic functions. Pacific J. Math 9, 975–985 (1959) 13. F. Bayart, A. Mouze, Division et composition dans l’anneau des séries de Dirichlet analytiques. Ann. Inst. Fourier 53(7), 2039–2060 (2003) 14. J.P. Kahane, H. Queffélec Ordre, convergence et sommabilité de produits de séries de Dirichlet. Ann. Inst. Fourier, Grenoble 47(2), 485–529 (1997) 15. B. Maurizi, Construction of an ordinary Dirichlet series with convergence beyond the Bohr strip. Missouri J. Math. Sci. 25(2), 110–133 (2013) 16. H. Bohr, Über die gleichmässige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913) 17. H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series. Ann. Math. 2, 600–622 (1931) 18. E. Hewitt, J.H. Williamson, Note on absolutely convergent Dirichlet series. Proc. Amer. Math. Soc. 8, 863–868 (1957) 19. E. Landau, Über den Wertevorrat von ζ(s) in der Halbebene σ>1, Nachrichten von der ges. der Wiss. Göttingen 1933, Fachgruppe I (Mathematik) Nr. 36, 81–91 20. L. Navas, Analytic continuation of the Fibonacci Dirichlet series. Fibonacci Quart. 39, 409–418 (2001)

Chapter 5

Probabilistic Methods for Dirichlet Series

5.1 Introduction The title of this chapter is a little emphatic, because the probabilistic methods will here concentrate essentially on one maximal inequality, which is fairly well-known in harmonic analysis, but will have a specific aspect, due to the Bohr point of view on Dirichlet series. We tried to keep the presentation as self-contained as possible, since the subject may be not completely familiar to some number-theorists. Let us emphasize that those probabilistic methods have great flexibility, and are nearly compulsory in some questions, even if the initial proof of the Bohnenblust–Hille theorem, to be proved in the last section, made no use of such methods. We first need a multidimensional version of the classical Bernstein inequality. We equip once and for all Rr with its sup-norm. That is if t = (t1 , . . . , tr ) ∈ Rr , we set: t = t∞ = max |t j |. 1≤ j≤r

The dual norm is the 1 -norm (used once in a proof): t1 =

r 

|t j |.

j=1

5.2 A Multidimensional Bernstein Inequality It will be convenient to know that the sup-norm supt∈[0,2π]r |P(t)| of a trigonometric polynomial P of given degree d in r variables can be estimated with a good accuracy from a limited number of values of t, say for t ∈ F, where the finite set F does not depend on P, but only on d and r . To that effect, we first give an estimate on © Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_5

129

130

5 Probabilistic Methods for Dirichlet Series

partial derivatives of P, which proves that P does not oscillate too violently, and in particular remains close to its maximal value for a “long” time. Theorem 5.2.1 (Multidimensional Bernstein inequality) Let 

P(t1 , . . . , tr ) = P(t) =

cα exp [i α, t] ,

α=(α1 ,...,αr )∈Zr

 where α, t := rj=1 t j α j , be a trigonometric polynomial in r variables, with degree   r |α | ; c  = 0 . Then, we have for all s, t ∈ Rn : d := max j α j=1  r    ∂P  π π    ∂x (t) ≤ 2 d P∞ and |P (s) − P (t)| ≤ 2 d t − s P∞ . j j=1 Proof We will need the following classical lemma ([1], p. 207): Lemma 5.2.2 If z 1 , . . . , zr are complex numbers, there exist real signs ε1 , . . . , εr with ε j = ±1, such that r r  2    εjz j ≥ |z j |. (5.2.1)  π j=1 j=1 Now, fix t ∈ Rr and consider the polynomial in one variable given by Q(u) = P(t1 + ε1 u, . . . , tr + εr u) =



r  cα exp (i t, α) exp[i( ε j α j )u]. j=1

Clearly, the degree of Q is ≤ d, so the one-dimensional case of Bernstein’s inequality ([1], p. 214) gives |Q (0)| ≤ dQ∞ ≤ dP∞ . That is, r  ∂ P   εj (t) ≤ dP∞ .  ∂x j j=1 ∂P Choosing the ε j as in the lemma, with z j = ∂x (t), we get the first part of j Theorem 5.2.1. For the second part, apply the mean-value theorem (in which D denotes the differential, whose norm is the dual norm on Rr , namely, the 1 -norm):

|P(s) − P(t)| ≤ s − t∞ sup D P(w) = s − t∞ sup w∈[s, t]

 r    ∂P   . (w)  ∂x 

w∈[s,t] j=1

We now conclude with the freshly proved Bernstein inequality.

j



5.2 A Multidimensional Bernstein Inequality

131

Remark The main aspect of this inequality is that it is “dimension-free”, if we ignore the new factor π2 > 1 (by the way, we ignore if we can completely get rid of such a factor), in that the parameter r does not appear in the majorizations, but only the parameter d. The following corollary will be of essential importance for us. It refers to what is sometimes called “arithmetic diameter” (see [1], p. 545). Corollary 5.2.3 Let d and r be positive integers. Then, there exists a finite set F ⊂ [−π, π]r , with cardinality |F| ≤ (Kd)r (K being a numerical constant), such that, for any trigonometric polynomial P of degree d in r variables, we have P∞ ≤ 2 sup |P(t)|.

(5.2.2)

t∈F

Proof Let ε = 1/π 2 d and B be the closed unit ball of Rr . According to a simple and well-known fact in local Banach space geometry ([1], p. 343), there exists an ε-net R ⊂ B of cardinality |R| ≤ (1 + 2/ε)r (i.e. dist (u, R) ≤ ε for any u ∈ B). Now, F := π R is a πε = 1/πd-net of [−π, π]r , with cardinality |F| = |R| ≤ (1 + 20d)r ≤ (21d)r =: (K d)r . Let P be as in the statement, M = sups∈F |P(s)| and t ∈ [−π, π]r . There is some s ∈ F such that s − t ≤ πε. Using Theorem 5.2.1, we have |P(t)| ≤ |P(s)| + |P(t) − P(s)| ≤ M +

1 π dt − s P∞ ≤ M + P∞ . 2 2

Passing to the supremum in t, we get P∞ ≤ M + 21 P∞ , that is P∞ ≤ 2M.  First, we emphasize a well-known, but important fact. Recall that D is the open unit disk in C and that T = {z; |z| = 1} is its boundary: r

Theorem 5.2.4 Let f be a function which is analytic in Dr and continuous on D . Then: (5.2.3) sup | f (z)| = sup | f (z)|. r

z∈D

z∈Tr

The common value of those two supremums is denoted by  f ∞ . Theorem 5.2.4 is called the “distinguished maximum principle”, because it asserts that f takes its maximum not only on the boundary of Dr (|z j | = 1 for some j) but also on its distinguished boundary (|z j | = 1 for all j). The proof is immediate by induction on the number of variables. We will indeed use the following form of Corollary 5.2.3. If n ∈ N, if α = (α1 , . . . , αr ) ∈ Nr and z = (z 1 , . . . , zr ) ∈ Cr , we use the multinomial notation: z α = z 1α1 · · · zrαr .

132

5 Probabilistic Methods for Dirichlet Series

Corollary 5.2.5 Let d and r be positive integers. Then, there exists a finite set H ⊂ Tr , with cardinality |H | ≤ (K d)r (Kbeing a numerical constant), such that, for any trigonometric polynomial P(z) = α∈Nr cα z α of degree d in r variables, we have (5.2.4) P∞ ≤ 2 sup |P (z) |. z∈H

Proof Just apply Corollary 5.2.3 to the trigonometric polynomial   Q (t1 , . . . , tr ) = P eit1 , . . . , eitr and the finite set H =

  it

e 1 , . . . , eitr ; t = (t1 , . . . , tr ) ∈ F .



5.3 Random Polynomials 5.3.1 Maximal Functions in Probability Let (, A, P) be a probability space, and let (X t )t∈F be a collection of non-negative, random variables indexed by a finite set F with N ≥ 2 elements, and with moments of every order. Let M = supt∈F X t be the associated maximal function. We want to estimate the expectation E(M) in a fairly accurate way, which is a difficult task. This is possible if the tails of the X t ’s are uniformly small, in a sub-Gaussian sense expressed on their moments: √ Theorem 5.3.1 Suppose that X t  p ≤ C PX t 2 for each t ∈ F and for each 1 ≤ p < ∞, and some constant C > 0. Then (remembering that N = |F|):  E (M) ≤ C 2e log N sup X t 2 .

(5.3.1)

t∈F

Proof We majorize M slightly better than by M≤



p

Xt

1p

 t∈F

X t , writing

, 1 ≤ p < ∞.

t∈F

Let σ = supt∈F X t 2 . By Hölder, we get:   p     p p  p E X t2 2 ≤ C p p 2 N σ p . E Xt ≤ C p p 2 [E (M)] p ≤ E M p ≤ t∈F

Taking pth roots, this gives: E (M) ≤ C p = 2 log N ≥ 1 and we get (5.3.1).

t∈F

√

1

pN p σ. Optimizing gives the value 

5.3 Random Polynomials

133

5.3.2 The Sub-Gaussian Aspect of Rademacher-Type Variables In order to apply Theorem 5.3.1, we first need a lemma. Lemma 5.3.2 Let X be a random variable which is both centered and of modulus bounded by 1. Then:   λ2 E eλX ≤ e 2 for all λ ∈ R. (5.3.2) Proof The function x → eλx is convex, and therefore below its chord x → chλ + x shλ for −1 ≤ x ≤ 1. In particular, eλX ≤ chλ + X shλ.   λ2 Integrating, we get E eλX ≤ chλ ≤ e 2 .



Now, the sub-Gaussian character of sums of bounded independent variables is best expressed by the Theorem 5.3.3 (Khintchine inequalities) Let ε be a standard Rademacher variable, i.e.   1 P (ε = 1) = P (ε = −1) = , implying E (ε) = 0, E ε2 = 1. 2 standard Rademacher Let ε1 , . . . , ε N be independent copies of ε (a so-called  sequence), a1 , . . . , a N be complex numbers, and S = Nj=1 a j ε j . Then, we have S p ≤ C with C =

√

√

pS2 for all 1 ≤ p < ∞,

2e.

 Proof We can assume that p > 2, and that S22 = |a j |2 = 1. We first take the a j s real. Then, from Lemma 5.3.2 and independence, we get (using the symmetry of S):   ∞ N N a 2j λ2        λ2q E S 2q λ2 = E eλS = E eλa j ε j ≤ e 2 =e2. (2q)! q=0 j=1 j=1 In particular, if we fix q ≥ 1, we obtain   λ2 λ2 E S 2q ≤ λ−2q (2q)!e 2 ≤ λ−2q (2q)2q e 2 . √ √ We optimize by taking λ = 2q, which gives S2q ≤ 2eq. Now, if p > 2 and 2(q − 1) ≤ p < 2q where q ≥ 2 is an integer, we obtain Sq ≤ S2q ≤

   2eq ≤ e ( p + 2) ≤ 2ep.

134

5 Probabilistic Methods for Dirichlet Series

  If a j = u j + iv j , write S = U + i V with U = u j ε j and V = v j ε j . The Minkowski inequality in L p/2 then gives, since |S|2 = |U |2 + |V |2 :  1/2  1/2 1/2 = U 2p + V 2p S p = U 2 +V 2  p/2 ≤ U 2  p/2 + V 2  p/2 1/2 √  √ ≤ C p U 22 + V 22 = C pS2 . The best constants are thus the same for real or complex coefficients. This ends the proof of Theorem 5.3.3.  Remark The reader will find a sharper inequality (with C = 1) in ([1], p. 28), with a slightly more technical proof, based on a brute force calculation, which succeeds thanks to the independence and to the multinomial formula; it was by the way the initial proof of S. Bernstein. But in the sequel the exact value of the constant will be unimportant for us.

5.3.3 The Kahane Bound for Random Trigonometric Polynomials In this subsection, we will establish an upper estimate for random trigonometric polynomials, due to Salem and Zygmund [2] for the one-dimensional case. But here we need a multidimensional extension due to Kahane ([3], pp. 68–69), which will be more suited for Dirichlet series. We fix an integer r ≥ 2 and denote by α = (α1 , . . . , αr ) an element of Zr and by (εα )α∈Zr a collection of independent Rademacher variables defined on some probability space (, A, P). With those notations, we have Theorem 5.3.4 (Kahane) Let P (t1 , . . . , tr ) = P (t) =



cα exp [i α, t]

α∈Zr

be a trigonometric polynomialof degree d ≥ 2 in r variables (by definition, we have  r d = max j=1 |α j |; cα  = 0 ). Let Pω (t) =



εα (ω)cα exp [i α, t] , ω ∈ .

α∈Zr

Then (C being some numerical constant): E (Pω ∞ ) ≤ C

 α∈Zr

|cα |2

1/2  r log d.

(5.3.3)

5.3 Random Polynomials

135

Proof Let us set [P] = E (||Pω ||∞ ) to emphasize that [ ] is a norm on the set of trigonometric polynomials, the so-called Pisier norm on the set of almost surely continuous Fourier series ([1], Chap. 13). Now, let F be as in Corollary 5.2.3, with |F| = N ≤ (Kd)r , as well as X t (ω) √ = |Pω (t)|. Those variables satisfy the assumptions of Theorem 5.3.1 with C = 2e in view of the Khintchine inequalities. Therefore, an application of this theorem and of Corollary 5.2.3 gives us [P] ≤ 2E(sup X t ) ≤ 4e

 log |F| sup X t 2 .

t∈F

t∈F

But log |F| ≤ r log K d ≤ Cr log d (where C can change from a formula to another) and   2 |cα |2 |eiα,t | = |cα |2 X t 22 = α

α

does not depend on t. This ends the proof of Theorem 5.3.4.



5.3.4 Random Dirichlet Polynomials We now exploit Theorem 5.3.4 in the framework of Dirichlet series, with help of the Bohr point of view. We will first need some notations and definitions, borrowed from analytic number theory. + (1) If n is an integer  N ≥ 2,−sP (n) denotes the largest prime divisor of n. (2) If f (s) = n=1 an n is a Dirichlet polynomial, the associated randomized polynomial is N  εn (ω)an n −s , ω ∈ . f ω (s) = n=1

(3) π(y) denotes the number of primes ≤ y. The prime number theorem asserts that π(y) ∼

y as y → ∞. log y

(4) ω(n) denotes the number of distinct prime factors of n and (n) the number of prime factors of n counted with their multiplicity, i.e. n = p1α1 · · · prαr with α j > 0 =⇒ ω(n) = r and (n) = α1 + · · · + αr . The basic theorem of this section, to be used repeatedly, and which among other facts will provide the Bohnenblust–Hille answer to Bohr’s question is the following:

136

5 Probabilistic Methods for Dirichlet Series

Theorem 5.3.5 Let y ≥ 2 and N ≥ 3 be integers, and let f (s) = Dirichlet polynomial such that

Then, we have

where  f 2 =

 N n=1

N n=1

an n −s be a

an = 0 =⇒ P + (n) ≤ y.

(5.3.4)

 E ( f ω ∞ ) ≤ C f 2 π(y) loglog N

(5.3.5)

|an |2

1/2 .

Proof Let r = π(y), so that pr ≤ y < pr +1 and that an = 0 =⇒ n = p1α1 · · · prαr =: p α , with α = α (n) = (α1 , . . . , αr ) ∈ Nr . Let P =  f be the associated algebraic polynomial according to Bohr’s point of view, namely,   an z α(n) =: cα z α . P(z) = α∈Nr

α∈Nr

Note, with obvious notations, the commutation relation ( f )ω = ( f ω ) = Pω . If an = 0, we have N ≥ n = p1α1 · · · prαr ≥ 2α1 +···+αr so that α1 + · · · + αr ≤

log N . log 2

This shows that P is ofdegree d ≤ log N / log 2 and depends on r = π(y) variables. N Moreover, |cα |2 = n=1 |an |2 . We now obtain from Bohr’s transference principle and from Theorem 5.3.4, that (C changing from a formula to another):



1/2  r log d E ( f ω ∞ ) = E Pω ∞ ≤ C |cα |2 ≤C

N 

|an |2

1/2  π(y) loglog N .

n=1

This ends the proof of Theorem 5.3.5.



Remark If one wishes estimates on random polynomials which are free of the constraint P + (n) ≤ y, one can appeal to more involved probabilistic techniques. First, according to a very general result of Marcus and Pisier [4], one can replace without any loss in the estimates Rademacher variables by Gaussian ones. Then, one can use the tools attached to Gaussian processes, like the lemma of Slépian. This was initiated in [5] with the following estimate:

5.3 Random Polynomials

137

Theorem 5.3.6 Let 0 ≤ σ < 1/2. Then, one has N   N 1−σ   . εn n −σ−it  ≈ E sup  log N t∈R n=1

The lower bound is an obvious consequence of Bohr’s inequality and of the Prime Number Theorem. The upper bound is more difficult. This result was generalized by Lifshits and Weber [6]. Also, Weber [7] considered more general coefficients, sub-multiplicative ones, namely, he considered random polynomials of the form N 

εn dn n −s

n=1

where dn ≥ 0 and dmn ≤ dm dn if (m, n) = 1. We refer to [7] for more details and references.

5.4 The Proof of Bohnenblust–Hille’s Theorem 5.4.1 An Elementary Version In this subsection, we will see how Theorem 5.3.5 can be used to obtain a straightforward proof of the Bohnenblust–Hille theorem, by probabilistic means, and with no more knowledge in number theory than the prime number theorem under the Tchebychev form: y . π(y) ≈ log y Recall that the Bohr constant T is defined by T := sup{σa ( f ) −σu ( f ) , over all possible Dirichlet series f (s) =

∞ 

an n −s .}

n=1

It is easy to see (cf. Chap. 6) that 0 ≤ T ≤ 1/2, but the exact value of T , even its positivity, is not clear. It will be to denote by P N the vector space of  Nconvenient an n −s (which, from now on, we denote by P Dirichlet polynomials P(s) = n=1 rather than by f ), and to equip it with the two norms PW =

N  n=1

|an |; P∞ = sup |P(s)|. s∈C0

138

5 Probabilistic Methods for Dirichlet Series

We now have [8] a reformulation of the definition of T in finite terms: Lemma 5.4.1 We can as well define T by the formula T = inf{σ ≥ 0; PW ≤ Cσ N σ P∞ for all P ∈ P N , N ≥ 1}.

(5.4.1)

The proof of this Lemma 5.4.1 will be more conveniently done in Chap. 6, when some properties of the Banach algebra H∞ have been established. Now, we have the Theorem 5.4.2 (Bohnenblust-Hille) The exact value of T is 1/2. Proof We fix positive integers N , d and set y = N 1/d , r = π(y). We denote by A the set of square-free integers (namely ω(n) = (n)) obtained from p1 , . . . , pr and such that (n) = d, i.e.: n ∈ A ⇐⇒ n = pi1 · · · pid , with 1 ≤ i 1 < · · · < i d ≤ r. n ∈ A and an = 0 if n ∈ / A. Observe that n ∈ A =⇒ n ≤ y d ≤ N , so Set an = 1 if N −s that P (s) = n=1 an n ∈ P N . We now test (5.4.1) on the randomization Pω of P, for some admissible σ. Since Pω W = PW = |A|, we get |A| ≤ Cσ N σ Pω ∞ .

(5.4.2)

We now integrate (5.4.2) with respect to ω and use Theorem 5.3.5 to obtain  |A| ≤ Cσ N σ |A|1/2 r loglog N , or

 |A|1/2 ≤ Cσ N σ r loglog N .

But r ≤ y and

(5.4.3)

  N rd r ∼ δd |A| = ∼ d d! (log N )d

by the prime number theorem in the Tchebycheff form, so that (5.4.3) writes as well:

N 2 (log N )−d/2 ≤ Cσ,d N σ+ 2d 1

1

 loglog N .

Letting N tend to infinity with d fixed, we clearly get: 1/2 ≤ σ + 1/2d. Letting now d tend to infinity, we finally get σ ≥ 1/2 and T = 1/2, which ends the proof of the Bohnenblust–Hille theorem.  Remark The initial proof [9] was not probabilistic, but based on multilinear forms, heralding the Rudin–Shapiro sequence. It also used the Bohr transference principle. A modern, probability-free, presentation, with new results, can be found in [8].

5.4 The Proof of Bohnenblust–Hille’s Theorem

139

The “enemy” in using Theorem 5.3.4 is the number r of variables, which intervenes through its square root, while the degree d intervenes through its logarithm and plays a sluggish role. Correspondingly, the “enemy” in using Theorem 5.3.5 is y, the number of primes used, whereas N intervenes through its iterated logarithm and plays essentially no role. Therefore, everything is done to use only small prime factors, with at the same time a big number of integers. This is possible (Rankin’s problem). We shall obtain precise upper bounds in Chap. 6, when we estimate in a fairly accurate way the Sidon constant of the set {log 1, . . . , log N }, showing in particular that no ultraflat Dirichlet polynomials exist, contrary to the case of trigonometric polynomials ([10] and also [11]). Here, we obtain a pretty sharp lower bound.

5.4.2 A Sharp Version of the Bohnenblust–Hille Theorem The new definition of T poses more generally the question of the behaviour of the so-called Sidon constant of the set d ,  N = {log 1, log 2, . . . , log N } ⊂ R = R (where R is the Bohr compactification of R), i.e. of the best constant S( N ) such that ||P||W ≤ S ( N ) ||P||∞ for all P ∈ P N . In more explicit terms, S ( N ) is the best constant such that N  n=1

N     |an | ≤ S( N ) an n it  for any scalars a1 , . . . , a N . n=1

∞

(5.4.4)

We have the following lower bound for S( N ), which gives much more than the Bohnenblust–Hille theorem, and will turn out to be sharp, which was a surprise for the authors, because it shows that here probabilistic methods capture the essence of the problem. For large N , it will be convenient to set λ (N ) =



log N log2 N .

(5.4.5)

where log2 = loglog is the second iterate of the logarithm. With those notations, we have the following result, due to Régis de la Bretèche [12]. A previous lower bound (using the same tools) had been previously obtained (as well as the corresponding √ upper √ bound) by Konyagin and Queffélec [13], but with the constant 2 instead of 1/ 2, due to a less clever choice of the forthcoming parameter y.

140

5 Probabilistic Methods for Dirichlet Series

Theorem 5.4.3 We have for S( N ) the lower bound:   √ √ S( N ) ≥ a N exp −(1/ 2 + o(1))λ(N )

(5.4.6)

(where a is a positive constant). Proof We will use a combination of analytic number theory and probability. We recall that P + (n) is the largest prime divisor of the integer n and we define S(x, y) and (x, y) by: S (x, y) = {n ≤ x; P + (n) ≤ y},  (x, y) = |S(x, y)|.

(5.4.7)

If ρ is the Dickman function ([14], p. 370), setting u = log x/ log y, we will use the uniform estimate  (x, y) ∼ xρ(u) (5.4.8) in the domain (for fixed ε > 0)   5 exp (log2 x) 3 +ε ≤ y ≤ x, x ≥ 3,

(5.4.9)

as well as the estimate ([14], p. 377)   ρ(u) = exp −u(log u + loglog(2u) + O(1)) .

(5.4.10)

Now, taking x = N , we test the definition of S( N ) against the polynomial P(s) =



n −s and its randomization Pω (s) =

n∈S(N ,y)



εn (ω)n −s ,

n∈S(N ,y)

√ which verifies Pω W = (N , y) and P2 = (N , y). Integrating the inequality Pω W ≤ S( N )Pω ∞ and using the basic estimate (5.3.5) of Theorem 5.3.5, we obtain (majorizing π(y) by y):   (N , y) ≤ C S( N ) (N , y) y log2 N , that is  −1 (N , y) S( N ) ≥ C . y log2 N

(5.4.11)

Now, for some constant α > 0 to be adjusted, we choose y = exp[αλ(N )], which is in the domain of validity of (5.4.9). An easy computation gives  1 u= α

log N log2 N

and u log u =

(1 + o(1)) λ(N ). 2α

5.4 The Proof of Bohnenblust–Hille’s Theorem

141

Finally use the relation (5.4.11), remembering that (N , y) ∼ N ρ(u) and use (5.4.10) which gives log ρ(u) = −(1 + o(1))u log u, as well as the previous estimate of u log u. We obtain     N ρ(u) N log ρ(u) log y S( N )  = exp − log2 N y log2 N 2 2   

1 α N exp − + + o(1) λ(N ) . = log2 N 4α 2 

√ The optimal choice α = 1/ 2 now gives (5.4.6).



Let us mention that, in [13], the following was proved: Theorem 5.4.4 We have for S( N ) the upper bound:   √ S( N ) ≤ a N exp − bλ(N )

(5.4.12)

(where a and b are positive constants). This is in strong contrast with the case of the set E N = {1, 2, . . . , N } ⊂ Z =  T, √ the Sidon constant S(E N ) of which is known to verify S (E N ) ∼ N , for example through the existence of “ultraflat trigonometric polynomials” (see e.g. [3], p. 75). √ Also mention that de la Bretèche [12] found for b the explicit value b = 1/ 2 2 . We will return to this improved, and close to optimal, upper bound of S( N ) in Chap. 6. In the following exercises, we take the liberty of using some definitions and facts of Chap. 6, which constitutes the main application and justification of random methods.

5.5 Exercises 1. Let 0 < α < 21 . (i) Show that there exists a Dirichlet series f such that σa ( f ) =

1 and σu ( f ) = 0. 2

142

5 Probabilistic Methods for Dirichlet Series

(ii) Let h α (s) = ζ(s +

1 2

+ α) and g = f + h α . Show that σa (g) =

1 1 and σu (g) = − α. 2 2

This is the proof, due to Bohr, that the difference σa (g) − σu (g) can take any value α such that 0 ≤ α ≤ 21 . 2. Fix an integer d ≥ 1 and denote by H∞ (d) the Banach subspace of H∞ formed −s ∈ H∞ such that by Dirichlet series f (s) = ∞ n=1 an n an = 0 =⇒ (n) ≤ d. Let us set

Td = sup{σa (g) − σu (g); g ∈ H∞ (d)}.

(i) Show that Td = inf{σ ≥ 0; PW ≤ Cσ N σ P∞ for all P ∈ P N (d)}, where P N (d) denotes the set of Dirichlet polynomials P(s) =

N 

an n −s with (n) > d =⇒ an = 0.

n=1

(ii) By using the Littlewood–Blei inequality for multilinear forms, show that Td ≤ 1/2 − 1/2d. (iii) By using the probabilistic methods of this chapter, show that Td ≥ 1/2 − 1/2d and finally that Td = 1/2 − 1/2d. Why does that give a new proof of the Bohnenblust–Hille theorem? 3. Consider the random Dirichlet series f ω (s) =

∞  εn (ω) −s n √ n log n n=2

where (εn ) is a Rademacher sequence. (i) Show that we have σc ( f ω ) = 0 almost surely.  −s with the following (ii) Show that there exists a Dirichlet series f (s) = ∞ n=1 an n properties: ∞  n=1

|an |2 < ∞; σc ( f ) = 0; {s; Re s = 0} is a natural boundary for f.

5.5 Exercises

143

(iii) Could you produce an example that does not use the random method of (ii)? 4. Let (εn ) be a Rademacher sequence. We admit here the following theorem of  −λn s a e is a general Dirichlet series with conSteinhaus ([3], p. 44): if f (s) = ∞ n n=1 ∞ vergence abscissa 0, then the series n=1 εn an e−λn s has almost surely the imaginary axis Re s = 0 as its natural boundary.  −s such that Show that there is a Dirichlet series f (s) = ∞ n=1 an n (i) |an | is non-increasing; (ii) f and all its derivatives belong to H∞ ; (iii) The line {Re s = 0} is a natural boundary for f .

References 1. D. Li, H. Queffélec, Introduction à l’étude des espaces de Banach, Analyse et probabilités, Cours spécialisés SMF, N o 12, (2004) 2. R. Salem, A. Zygmund, Some properties of trigonometric series whose terms have random signs. Acta Math. 91, 245–301 (1954) 3. J.P. Kahane, Some Random Series of Functions, 2nd edn. (Cambridge University Press, 1985) 4. M. Marcus, G. Pisier, Characterization of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152, 245–301 (1984) 5. H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995) 6. M. Lifshits, M. Weber, On the supremum of random Dirichlet polynomials, preprint. 7. M. Weber, supremum of random Dirichlet polynomials with sub-multiplicative coefficients. Unif. Distrib. Theory 5(1), 163–197 (2010) 8. B. Maurizi, H. Queffélec, Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 676–692 (2010) 9. H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series. Ann. Math. 2, 600–622 (1931) 10. J.P. Kahane, Sur les polynômes à coefficients unimodulaires. Bull. Lond. Math. Soc. 12, 321–342 (1980) 11. H. Queffélec, B. Saffari, On Bernstein’s inequality and Kahane’s ultraflat polynomials. J. Fourier Anal. Appl. 2, 519–582 (1996) 12. R. de la Bretèche, Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arith. 134(2), 141–148 (2008) 13. S. Konyagin, H. Queffélec, The translation 21 in the theory of Dirichlet series. Real Anal. Exch. (27), 155–175 (2001–2002) 14. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. (Institut Elie Cartan, 1995)

Chapter 6

Hardy Spaces of Dirichlet Series

6.1 Definition and First Properties 6.1.1 The Origin of the Spaces H∞ and H2 The forthcoming spaces H p of Dirichlet series (1 ≤ p ≤ ∞), analogous to the familiar Hardy spaces H p on the unit disk, have been successfully introduced to study completeness problems in Hilbert spaces [1], first for p = 2, ∞. Later on, the general case was considered in [2] for the study of composition operators. We will return to that general case further in this chapter, and now concentrate on the cases p = 2, ∞. Here is the initial motivation: let H = L 2 (0, 1) and ϕ ∈ H , viewed as a 2-periodic odd function on R through its Fourier expansion ϕ(x) =

∞ 

∞  √ an 2 sin nπx, |an |2 < ∞.

n=1

(6.1.1)

n=1

Now, let ϕn ∈ H be the dilated function of ϕ defined by ϕn (x) = ϕ(nx). We observe √ that a natural orthonormal basis of H is formed by the sequence of functions 2 sin nπx, n = 1, . . ., which are the dilates of the single function ψ (x) = √ 2 sin πx. And we want to know for which functions ϕ ∈ H this still holds: exact orthonormality is surely too much to be required, because then the only possibility is easily seen to be the function ψ! Therefore we weaken a little our ambitions and ask under which conditions the system (ϕn )n≥1 of those dilated functions verifies either of the conditions: 1. The system (ϕn ) is complete, i.e. the space generated by the ϕn is dense in H . 2. The system (ϕn ) is complete and, for some constant C, we have: C

−1

 ∞ n=1

|cn |

2

 21

   21  ∞   ∞ 2   ≤ cn ϕn  ≤ C |cn | n=1

2

(6.1.2)

n=1

© Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_6

145

146

6 Hardy Spaces of Dirichlet Series

where all but a finite number of cn are non-zero. One then says that (ϕn ) is a Riesz basis in H (the image of an orthonormal basis under an onto isomorphism). Surprisingly (cf. Theorem 6.4.1 to come), the (complete) answer to Question 2 will be in terms of the forthcoming spaces of Dirichlet series, which had been suggested by Beurling as early as in 1945. Recall that Cθ = {s ∈ C; Re s > θ}. We will denote by D the set of series ∞ −s n=1 an n such that σc < ∞ (the set of convergent Dirichlet series), equivalently −s whose coefficients have at most polynomial growth. We will of series ∞ n=1 an n 2 denote by H the space of Dirichlet series with square-summable coefficients: f ∈ H2 ⇐⇒ f (s) =

∞ 

an n −s , with

n=1

∞ 

|an |2 =: f 22 < ∞.

(6.1.3)

n=1

And we will denote by H ∞ (C0 ) the set of bounded analytic functions on C0 , equipped with the norm f ∞ = sups∈C0 | f (s)|. Now, the set H∞ of bounded Dirichlet series (later seen to be the set of multipliers of H2 ) is by definition: H∞ = H ∞ (C0 ) ∩ D.

(6.1.4)

In other terms, a function f belongs to H∞ if it verifies two properties: (i) It is a bounded analytic function in C0 . (ii) Moreover, it can be represented as a convergent Dirichlet series for Re s large enough. Let us examine some examples: ∈ / H∞ . Indeed, f is analytic and bounded by 1 in C0 , but f ∈ / D. (1) f (s) = s−1 s+1 −s / H∞ , for the same reasons. (2) f (s) = e ∈  n−1 −s−1 n ∈ / H∞ . We have well the (3) f (s) = (1 − 2−s )ζ(s + 1) = ∞ n=1 (−1) membership of f in D, but f is not bounded in C0 , for example because of Bohr’s inequality and of the fact that p 1p = ∞.  n ∞ (4) Let g(z) = ∞ n=0 bn z ∈ H , the space of functions which are analytic and bounded in the open unit disk D, equipped with the norm g ∞ = supz∈D |g(z)|. Then, the mapping into

g → (g) = f, f (s) = g(2−s ), is an isometry : H ∞ −→ H∞ . This is obvious since g(s) =

∞ n=0

bn (2n )−s ∈ H∞ .

6.1.2 A Basic Property of H∞ To estimate the coefficients of a function in H∞ , we have the following starting point, of basic utility.

6.1 Definition and First Properties

Theorem 6.1.1 If f (s) =

∞ n=1

147

an n −s ∈ H∞ , then

|an | ≤ f ∞ , ∀ f ∈ H∞ , ∀n ∈ N∗ ; and σa ( f ) ≤ 1.

(6.1.5)

∞ −ρ < ∞. Such a ρ exists since Proof Let ρ > 0 such that n=1 |an |n σa ( f ) ≤ σc ( f ) + 1. We can then write the Fourier-Bohr formulas:

an n

−ρ

1 = lim T →∞ 2 T

equivalently 1 an = lim T →∞ 2i T

T f (ρ + it)n it dt, −T

ρ+i  T

f (s)n s ds. ρ−i T

By the Cauchy integral theorem for a rectangle, we have as well ⎞ ⎛ ρ+i T ρ−i ε+i   T  T 1 ⎝ f (s)n s ds + f (s)n s ds − f (s)n s ds ⎠ . an = lim T →∞ 2i T ε+i T

ε−i T

ε−i T

The first and third terms are dominated by ρn ρ f ∞ /2T and tend to 0 as T → ∞. The second term is majorized by n ε f ∞ , so that |an | ≤ f ∞ , by letting ε tend to  0. This implies that σa ( f ) ≤ 1.

6.2 The Banach Space H∞ 6.2.1 The Banach Algebra Structure of H∞ The following theorem gives some information on the structure of H∞ . Theorem 6.2.1 The space H∞ , equipped with the norm ∞ , is a unital and nonseparable Banach algebra. Its invertible elements f are characterized by the relation | f (s)| ≥ δ for some δ > 0.  ∞ −s −s Proof Let f (s) = ∞ ∈ H∞ and h = f g. The funcn=1 an n , g(s) = n=1 bn n tion h is analytic and bounded in C0 . Moreover, we have max(σa ( f ), σa (g)) ≤ 1 by Theorem therefore h is represented by the absolutely convergent Dirichlet  6.1.1, −s c n , with c = a ∗ b, for Re s > 1, showing that h ∈ H∞ , with clearly series ∞ n n=1 h ≤ f g ∞ . Let now ( f j ) be a Cauchy sequence of H∞ , with f j (s) = ∞∞ j −s ∞ ∞ n=1 an n , and C = sup j f j ∞ < ∞. Since the space H (C0 ) of bounded ana-

148

6 Hardy Spaces of Dirichlet Series

Fig. 6.1 Shifted vertical line

lytic functions in C0 is a Banach space for the ∞ -norm, there exists some f ∈ H ∞ (C0 ) such that lim j→∞ f j − f ∞ = 0 (Fig. 6.1). j It remains to prove that f ∈ H∞ . By Theorem 6.1.1, we have |an − ank | ≤ j j f j − f k ∞ , showing that lim j→∞ an = an exists for each n, with moreover |an | ≤ f j ∞ ≤ C and |an | ≤ C by passing to the limit. This shows that the Dirichlet series  ∞ −s is absolutely convergent for Re s > 1. Fix such an s with Re s = σ > 1 n=1 an n  j −s and observe that the series ∞ is normally convergent with respect to j since n=1 an n j −s −σ |an n | ≤ n . Therefore, we can permute limit and sum to get f (s) = lim

j→∞

∞  n=1

anj n −s =

∞  n=1

( lim anj )n −s = j→∞

∞ 

an n −s .

n=1

This proves that H∞ is a Banach algebra. Now, if f is invertible with inverse g, we have for each s ∈ C 0 : 1 = | f (s)g(s)| ≤ g ∞ | f (s)|, so that | f (s)| ≥ 1/ g ∞ . −s ∈ H∞ and | f (s)| ≥ δ > 0, we obviously have Conversely, if f (s) = ∞ n=1 an n ∞ we get |a1 | ≥ δ by letting Re s = σ tend to infinity. 1/ f ∈ H (C0 ). Moreover,  |an | −σ < 1, a Neumann type argument allows to expand Therefore, as soon as ∞ n=2 |a1 | n

6.2 The Banach Space H∞

149

  1 1 1  (s) = f a1 1 + ∞ n=2

an −s n a1

 −s ∞ as a convergent Dirichlet series ∞ n=1 bn n , showing that 1/ f ∈ H . It remains to show that H∞ is non-separable. We recall that the Banach space H ∞ of bounded, analytic functions on the unit disk D is non-separable. Indeed, if h a (z) = e−a[(1+z)/1−z] for a > 0, we have h a ∈ H ∞ and b = a =⇒ h a − h b ∞ = sup |h a (z) − h b (z)| = sup |e−as − e−bs | z∈D

s∈C0

= sup |e−ait − e−bit | = 2 t∈R into

by the maximum modulus principle. And the already mentioned map  : H ∞ −→ H∞ defined by g(s) = g(2−s ) is an isometry, so that H∞ is not separable either. This ends the proof of Theorem 6.2.1. 

6.2.2 Behaviour of Partial Sums The partial sums of a function in H∞ are well-controlled as shown by the following theorem [3]: Theorem Let us consider a function 6.2.2 (Balasubramanian–Calado–Queffélec) N −s ∞ −s f (s) = ∞ a n ∈ H and S (s) = a n , its partial sum of index N ≥ N n=1 n n=1 n 2. Then (6.2.1) S N ∞ ≤ C log N f ∞ where C is a numerical constant. Proof From Theorem 6.1.1, we know that |an | ≤ f ∞ , so that σa ( f ) ≤ 1. We can apply the Perron-Landau effective formula (Theorem 2.3 of Chap. 4) with ρ = 2 and x = N + 21 a half-integer, namely 

1 A(x) := an = 2iπ n≤x

2+i  T

2−i T

xs f (s) ds + O s



x 2  |an | . T n≥1 n 2 | log nx |

  We easily see that | log(x/n)| is ≥ 1/4 N + 21 if n > x and is ≥ 1/4N if n < x, so that the error term is dominated by C x 3 ( f ∞ /T ) = C f ∞ if we adjust T = x 3 , the constant C being absolute. Now, we shift the integral term using the Cauchy integral formula as follows (with 0 < ε < 2) (Fig. 6.2):

150

6 Hardy Spaces of Dirichlet Series

Fig. 6.2 Shifted vertical line

2+i  T

xs f (s) ds = s

2−i T

ε+i  T

ε−i T

xs f (s) ds + s

2 −

f (u − i T ) ε

2 f (u + i T ) ε

x u+i T du u + iT

x u−i T du. u − iT

The last two integrals are uniformly dominated by (x 2 /T ) f ∞ = f ∞ /x since T = x 3 . For the first one, we have   T  T  ε+it   xε x ≤  f (ε + it) idt f dt √ ∞   ε + it ε2 + t 2   −T

−T

ε

T /ε

= 2x f ∞

√ 0

du u2 + 1

6.2 The Banach Space H∞

151

⎛ ≤ 2x ε f ∞ ⎝1 +

T /ε

⎞ du ⎠ u

1

  T ε . ≤ 4x f ∞ log ε Now, we adjust ε = 1/ log x so that x ε = e, and we finally obtain (changing C if necessary):   |A (x)| ≤ C || f ||∞ log x 3 log x ≤ C || f ||∞ log x which ends the proof of Theorem 6.2.2. Indeed, if g(s) = f (s + s0 ) where s0 ∈ C0 , we have from the preceding  N    −s0   an n  ≤ C log N g ∞ ≤ C log N f ∞ ,  n=1

implying S N ∞ ≤ C log N f ∞ .



Remark The previous proof uses the Perron formulas and a little complex analysis. A simpler proof, using only harmonic and real analysis, was recently found by Saksman [4]. We come back to this in Sect. 6.4.3.

6.2.3 Some Applications of the Control of Partial Sums As a first consequence of Theorem 6.2.2, we have a simple proof of the following Theorem of Bohr [5]; indeed, the proof of Theorem 6.2.2 might be viewed as a quantitative version of Bohr’s Theorem). This theorem deals with uniform convergence in a whole half-plane and is of central importance in the study of H∞ [3]:  −s Theorem 6.2.3 (Bohr’s theorem) Let f (s) = ∞ ∈ H∞ , Then, the asson=1 an n ciated Dirichlet series converges uniformly in each half-plane Cε , ε > 0. In other terms, we have σu ( f ) ≤ 0, and in particular σc ( f ) ≤ 0. ∞ Proof We must show that n −s−ε converges uniformly in C0 for each n=1 an ε > 0. To that effect, we set Sn (s) = nj=1 a j j −s , S0 (s) = 0 and perform an Abel’s transformation: N 

an n −s−ε =

n=1

N 

[Sn (s) − Sn−1 (s)]n −ε

n=1

=

N −1  n=1

Sn (s)[n −ε − (n + 1)−ε ] + S N (s)N −ε .

152

6 Hardy Spaces of Dirichlet Series

Now, the series in the RHS is normally convergent in C0 , since its general term  is dominated by Cε log n/ n ε+1 f ∞ by Theorem 6.2.2, while the remaining term Sn (s)N −ε is dominated in C0 by C log N /N ε , which ends the proof.  An interesting aspect of that theorem is that, in the end, the Dirichlet series attached to a function f ∈ H∞ converges to that function where it is defined, namely in C0 ! Remark Bohr [5] actually proved his theorem on uniform convergence for more general Dirichlet series ∞ 

an e−λn s with λn+1 − λn ≥ δe−Aλn

n=1

where δ and A are positive constants. This condition holds with A = 1 for λn = log n (ordinary Dirichlet series) but fails to hold if for example λn = (log n)α with 0 < α < 1. This general version can be proved essentially in the same way as Theorem 6.2.2, using the elementary Perron-Landau formulas  of Chap. 4 with x = (λ N + λ N +1 )/2 and y = e x−λn , to get control of A∗ (x) = λn ≤x an . A little later, Landau [6] relaxed the condition on the sparseness of exponents to ∞ 

an e−λn s with λn+1 − λn ≥ δε exp(−eελn ) for all ε > 0.

n=1

This allows this time exponents like λn = (log n)α with 0 < α < 1. As a second consequence of Theorem 6.2.2, we can now extend Bohr’s inequality from Dirichlet polynomials to an arbitrary element of H∞ :  −s Theorem 6.2.4 (Bohr’s inequality) Let f (s) = ∞ ∈ H∞ . Then, we have n=1 an n 

|a p | ≤ f ∞ .

(6.2.2)

p

Proof If g ∈ H∞ and ε > 0, set (Tε g)(s) = g(s + ε). We have Tε g ∈ H∞ and Tε g ∞ ≤ g ∞ . Now, by the Bohr uniform convergence theorem, there is an integer N0 = N0 (ε) such that Tε S N ∞ ≤ Tε f ∞ + ε ≤ f ∞ + ε, ∀N ≥ N0 .

6.2 The Banach Space H∞

153

The Bohr inequality for the Dirichlet polynomial Tε S N therefore gives 

|a p | p −ε ≤ Tε S N ∞ ≤ || f ||∞ + ε.

p≤N

Let N → ∞ to get lemma to get (6.2.2).

 p

|a p | p −ε ≤ f ∞ + ε. Let now ε → 0 and use the Fatou 

Theorem 6.2.2 finally provides in the case of H∞ a very simple proof of an important theorem of Carlson, which will be used again in Chap. 7 under a more elaborate form (see also [7, 8]):  −s Theorem 6.2.5 (Carlson’s identity) Let f (s) = ∞ ∈ H∞ and ε > 0. n=1 an n Then, we have 1 lim T →∞ T

T | f (ε + it)|2 dt =

∞ 

|an |2 n −2ε ≤ f 2∞ .

(6.2.3)

n=1

0

In particular, we have the contractive inclusion H∞ ⊂ H2 , namely f ∈ H∞ =⇒ f ∈ H2 and f 2 ≤ f ∞ .  −ε −it converges uniformly on R Proof It suffices to notice that the series ∞ n=1 an n n  −iλn t b e , uniformly by Theorem 6.2.3, and that for a series of the type f (t) = ∞ n n=1 convergent on R, we always have (here with λn = log n) 1 lim T →∞ T

T | f (t)|2 dt = 0

∞ 

|bn |2 .

n=1

This can be checked by hand thanks to uniform convergence, or can be viewed as the Parseval formula for f in the Bohr compactification R of R. In any case, we have (6.2.3) and we get the second assertion by letting ε tend to 0. 

6.3 Additional Properties of H∞ Having established the most basic properties of the algebra H∞ , we now come to possibly more specialized, but yet quite striking, properties of that algebra of functions.

154

6 Hardy Spaces of Dirichlet Series

6.3.1 An Improved Montel Principle We have first the following theorem. Theorem 6.3.1 (Bayart) Let ( f j ) be a bounded sequence of H∞ . Then, we can extract from it a subsequence which converges uniformly on each half-plane Cε (and not only on compact subsets of C0 ).  ( j) −s ( j) Proof Write f j (s) = ∞ n=1 an n . We know that |an | ≤ f j ∞ ≤ C for some constant C. By the diagonal process, up to taking a subsequence, we may assume ( j) that lim j→∞ an = an exists for n = 1, 2, . . . and we have |an | ≤ C. Fix s ∈ C1 . The Weierstrass M-test implies that lim f j (s) =

j→∞

∞ 

de f

an n −s = f (s).

n=1

On the other hand, by Montel’s theorem on normal families of holomorphic functions, up to taking a subsequence again, we can assume that f j converges to some function g ∈ H ∞ (C0 ), uniformly on compact subsets of C0 . This implies f = g on C1 , so that f ∈ H∞ and that f = g on C0 by the uniqueness of holomorphic continuation. It remains to prove the uniform convergence of f j to f on half-planes Cε , equivalently that of g j (s) = f j (s + ε) − f (s + ε) to 0 in H∞ . To that effect, set  ( j) ( j) j n = nk=1 (ak − ak )k −s and 0 = 0. We see that, for s ∈ C0 , we have |g j (s)| ≤

N 

|an( j)

     ( j) ( j) −ε   − an | +  (n − n−1 )(s)n .

n=1

n>N

Now, since f j − f ∞ ≤ 2K , an Abel’s transformation and the use of Theorem 6.2.2 give g j ∞ ≤

N 

|an( j) − an | +

n=1

This implies lim sup g j ∞ ≤ j→∞

 2C K ε log n . n 1+ε n≥N

 2C K ε log n . n 1+ε n≥N

Letting N tend to ∞, we get the result.



Let us mention that Theorem 6.3.1 was used by F. Bayart [2] to give a full characterization of compact composition operators on H∞ .

6.3 Additional Properties of H∞

155

6.3.2 Interpolating Sequences of H∞ Finally, we would like to mention a last result in that section, which needs a definition: we say that a sequence (sn )n≥1 of points of C0 is an interpolation sequence for some algebra A of bounded functions defined on C0 if, for any bounded sequence (wn )n≥1 of complex numbers, there exists f ∈ A solving the following infinite Lagrange interpolation problem: f (sn ) = wn for all n ≥ 1. when the algebra A is that of all bounded holomorphic functions on C0 , the next result is a famous theorem of Carleson [9, p. 278]. Theorem 6.3.2 (Carleson) The sequence (sn ) ⊂ C0 is interpolating for H ∞ (C0 ) if and only if the following holds:     sn − s j     s + s  = δ > 0. n≥1 j j=n n

(6.3.1)

inf

Moreover, if (wn ) is a bounded sequence of scalars, there exists f ∈ H ∞ (C0 ) such that f (sn ) = wn for all n and C log(1 + 1/δ) sup |wn |. δ n

f ∞ ≤

This condition, generally stated on the unit disk and then transferred to a half-plane by conformal mapping, has the merit of being fairly explicit and can be efficiently tested on examples. If we replace the disk by the polydisk, only sufficient conditions are known. Let us quote the following result [10]: Theorem 6.3.3 (Berndtsson–Chang–Lin) Let dm be the Gleason distance in the polydisk Dm , namely for z = (z 1 , . . . , z m ), w = (w1 , . . . , wm ) in Dm :   z j − wj dm (z, w) = max  1≤ j≤m 1 − w z

j j

  . 

Let (z (n) ) be a sequence of distinct points of Dm . Then, a sufficient condition for that sequence to be interpolating for the Banach algebra H ∞ (Dm ) is that inf

n≥1



dm (z (n) , z ( j) ) = δ > 0.

j=n

Moreover, if (wn ) is a bounded sequence of scalars, there exists a function f ∈ H ∞ (Dm ) such that f (z (n) ) = wn for all n and f ∞ ≤

100 log(1 + 1/δ) sup |wn |. δ n

156

6 Hardy Spaces of Dirichlet Series

Let us now turn to the more difficult problem of interpolation in the smaller algebra H∞ . Using cleverly Theorem 6.3.3, and in a manner quite in the spirit of the algebra H∞ , K. Seip was able to obtain the following theorem. Theorem 6.3.4 (Seip) Let S = (sn ) be a bounded sequence of C0 . Then, this sequence is interpolating for H∞ if and only if it is interpolating for the bigger algebra H ∞ (C0 ). Proof We will use Theorem 6.3.3 for m = 2, and abbreviate dm to d. We will also set, for a, b ∈ C0 :   a − b   , and T (a) = (2−a , 3−a ) ∈ D2 . ρ(a, b) =  (6.3.2) a + b We assume that S is bounded and interpolating for H ∞ (C0 ). We first need a technical lemma. Lemma 6.3.5 Let M = supa∈S |a| and r = inf a,b∈S ρ(a, b) > 0. Then, there exists C = C(M) > 1 such that, if a, b ∈ S, we have

a=b

1 − d(T (a) , T (b)) ≤ C [1 − ρ(a, b)] , d(T (a), T (b)) ≥ C −1 ρ(a, b). (6.3.3) Moreover, there exists a constant μ = μ(r, M) > 0 such that a, b ∈ S and a = b =⇒ d[T (a), T (b)] ≥ [ρ(a, b)]μ .

(6.3.4)

Proof of the Lemma Write a = α + it, b = β + iτ . Denote by C > 1 a constant depending only on M, which can vary. For p = 2 and p = 3, we have   −a  p − p −b 2   1 − d [T (a) , T (b)] ≤ 1 −  1 − p −a−b     1 − p −2α 1 − p −2β 4αβ = ≤C 2 ) 2 (t−τ ) (α + β) + sin2 log p (t−τ 1 − p −(α+β) + 4 p −α−β sin2 log p 2 2 2

so that 1 − d 2 [T (a) , T (b)] ≤ C

  4αβ = C 1 − ρ2 (a, b) . 2 2 (α + β) + (t − τ )

We used the fact that the continuous function   2 sin x log 2 sin2 x log 3 f (x) = max , x2 x2 has a positive minimum on the compact interval [−M, M] since it does not vanish on that segment, due to the irrationality of log 2/ log 3. This proves the first

6.3 Additional Properties of H∞

157

half of (6.3.3), and the second one is proved similarly as follows: if we set d = d [T (a) , T (b)] and ρ = ρ (a, b), we have for p = 2, 3:    1 − p −2α 1 − p −2β 1 4αβ −1≤ ≤ C2 ,  2 ) 2 2  p −a − p −b  d (α − β) + sin2 log p (t−τ 2 so that 1 4αβ − 1 ≤ C2 = C2 2 d (α − β)2 + (t − τ )2



 1 C2 − 1 ≤ − 1, ρ2 ρ2

implying d ≥ C −1 ρ. Let us now turn to (6.3.4). According to (6.3.3), we have d ≥ C −1 ρ ≥ C −1r. Therefore (using log(1 − x)/x ≥ −λ for 0 ≤ x ≤ 1 − C −1r ), there exists some number λ = λ(M, r ) > 0 such that   d [T (a) , T (b)] =: 1 − ε (a, b) ≥ exp −λε (a, b) . Since moreover ρ(a, b) := 1 − ε(a, b) ≤ eε(a,b) , (6.3.3) implies   d [T (a) , T (b)] ≥ exp −λε (a, b) ≥ exp [−λCε(a, b)] ≥ [ρ (a, b)]λC , and μ = λC does the job.



As a corollary, we obtain the following proposition. Proposition 6.3.6 Let (sn ) ⊂ C0 be a bounded interpolation sequence for the algebra H ∞ (C0 ), and let T be as in (6.3.2). Then, the sequence (T (sn )) is an interpolation sequence for the Banach algebra H ∞ (D2 ). Proof We set r (a) =

 b∈S, b=a



ρ(a, b), r = inf r (a), τ (a) = a∈S

d[T (a), T (b)].

b∈S, b=a

According to Carleson’s interpolation theorem, we have r > 0. Then, fix a ∈ S. From (6.3.4), we obtain by pointwise multiplication: τ (a) ≥ [r (a)]μ ≥ r μ > 0. An appeal to Theorem 6.3.3 gives the conclusion.



Now, it is easy to prove Theorem 6.3.4. Indeed, suppose that (sn ) is interpolating for H ∞ (C0 ) and let (wn ) be a bounded sequence of scalars. According to

158

6 Hardy Spaces of Dirichlet Series

Proposition 6.3.6, we can find a function f ∈ H ∞ (D2 ) such that f [T (sn )] = wn for all n = 1, 2, . . . If F(s) = f [T (s)], using the Taylor expansion of f , we have F ∈ H∞ and moreover F(sn ) = f [T (sn )] = wn for all n = 1, 2, . . . 

This ends the proof.

6.4 The Hilbert Space H2 6.4.1 Definition and Utility In the introduction, we saw that a companion space to H∞ is the space H2 of Dirichlet series with square-summable coefficients: f ∈ H ⇐⇒ f (s) = 2

∞ 

an n

−s

with

n=1

f 22

=

∞ 

|an |2 .

n=1

It is easy to see that H2 is a Hilbert space with the scalar product  f, g =

∞ 

an bn

n=1

and that moreover it is a Hilbert space of analytic functions on the half-plane C1/2 (whereas H∞ is a space of analytic functions on the half-plane C0 !), with the natural orthonormal basis en , en (s) = n −s , n ≥ 1, so that its reproducing kernel K a , a ∈ C1/2 is given by K a (s) =

∞ 

en (s)en (a) = ζ(s + a) and

f (a) =  f, K a  ∀ f ∈ H2 .

n=1

We shall see later that H∞ is exactly the space of multipliers of H2 . Now, the answer to the initial question of this chapter is the following criterion [1], which combines the use of H2 and H∞ . Theorem 6.4.1 Let (ϕn ) be the system of dilated functions given by−s(6.1.1) and Sϕ ∈ H2 be the associated generating function, Sϕ(s) = ∞ n=1 an n . Then, the following are equivalent: (1) The dilated system (ϕn ) is a Riesz basis of H . (2) Both Sϕ and 1/Sϕ belong to H∞ .

6.4 The Hilbert Space H2

159

As an example of application, we see that if an = n −τ with Re τ > 1/2, the Re τ > 1. Indeed, Sϕ(s) = corresponding system (ϕn ) is a Riesz basis if and only if√ ζ(s + τ ) in this case. Observe that Sϕ = 1 for ϕ(x) = 2 sin πx. An immediate, but useful, property is the following proposition.  −s Proposition 6.4.2 Let f (s) = ∞ ∈ H2 . Then, σa ( f ) ≤ 1/2 and the result n=1 an n is optimal in general. Proof Just apply the Cauchy-Schwarz inequality: If Re s = σ > 1/2, then ∞ 

|an |n

−σ

≤

n=1

 ∞

|an | )

n=1

The function f (s) =

∞ n=2

2 1/2

 ∞

n

−2σ

1/2 < ∞.

n=1

√ n −s / n log n is in H2 and σa ( f ) = 1/2.



Here is a more refined result, due to Hedenmalm and Saksman [11].  −s Theorem 6.4.3 Let f (s) = ∞ ∈ H2 . Then, the series n=1 an n ∞ 

an n −1/2+it

n=1

converges for almost every real t. Proof We indicate (see [12]) a simpler proof than the original one, based on Carleson’s convergence theorem for integrals: let f : [1, ∞[→ C be defined by f (x) = an if n ≤ x < n + 1, n = 1, 2, . . . . ∞  2 x = e y shows Clearly, 1 | f (x)|2 d x = ∞ n=1 |an | . Now, the change of variable  ∞ that g : R+ → C defined by g(y) = f (e y )e y/2 is in L 2 (R+ ) with 0 |g(y)|2 dy = A ∞ 2 1 | f (x)| d x.Using Carleson’s theorem for integrals [13], we get that lim A→∞ 0 ∞ g(y)eit y dy =: 0 g(y)eit y dy exists for almost all t, equivalently that A lim

A→∞ 1

f (x) it log x d x exists for almost all t. √ e x

(6.4.1)

160

6 Hardy Spaces of Dirichlet Series

Now, N +1 

1

=

 f (x) it log x dx = √ e x n=1 N

n+1 N  



an

n=1 n

=:

N 

bn (t) +

n=1

n+1 it log x e an √ d x x n

eit log x eit log n √ − √ x n N 

 dx +

N 

an n − 2 +it 1

n=1

an n − 2 +it 1

n=1

with bn (t) = O(n −3/2 ), since an → 0 and ddx ( e √x ) = O(x −3/2 ). The integral  ∞ f (x) it log x  −1/2+it √ e d x and the series ∞ are thus simultaneously convern=1 an n 1 x gent and the result follows from (6.4.1).  it log x

6.4.2 The Embedding Theorem The usual Hardy-Hilbert space of the half-plane C1/2 , H 2 (C1/2 ), is defined as the space of functions f analytic in the half-plane C1/2 such that ∞

f 2H 2 (C1/2 )

:= sup

σ>1/2 −∞

| f (σ + it)|2 dt < ∞.

Those functions have almost everywhere radial limits f (1/2 + it) and the norm can be calculated by replacing σ by 1/2. This space is more flexible than H2 , and familiar to analysts, therefore the comparison of the two spaces, when available, sheds some light on H2 . In particular, the following theorem [14] turns out to be especially useful. Theorem 6.4.4 (Olsen–Seip) Let F ∈ H2 . Then, the function f defined by f (s) = F(s) is in H 2 (C1/2 ), and moreover s f H 2 (C1/2 )  F H2 .  Proof Without loss of generality, we can assume that F(s) = an n −s is a Dirichlet polynomial. Then, using in a crucial way the embedding theorem of Chap. 1 with √ an / n instead of an , we get:

6.4 The Hilbert Space H2



f 2H 2 (C1/2 )

161

−1 2  ∞     1 1 2   = dt  F 2 + it  4 + t −∞

k+1 −1 2      1  1 2 F  = dt + it + t   4 2 k∈Z k

k+1 2     1  2 −1     F 2 + it  (1 + k ) dt k∈Z k





(1 + k 2 )−1

 n

k∈Z

|an |2 



|an |2 ,

n



ending the proof.

Remark The embedding theorem and some variants play an important role ([15], see also Exercise 8) in the characterization of bounded composition operators on H2 which goes as follows. Let ϕ : C1/2 → C1/2 be an analytic function so that the map f → f ◦ ϕ =: Cϕ ( f ) (a “composition operator”) sends H2 to H (C1/2 ). Then: Theorem 6.4.5 (Gordon-Hedenmalm) The function ϕ determines a bounded composition operator Cϕ on H2 if and only if ϕ(s) = c0 s +

∞ 

cn n −s =: c0 s + ψ(s)

n=1

where c0 is a nonnegative integer and ψ is a Dirichlet series that converges in Cθ for some θ > 0 and has the following mapping properties: (1) If c0 ≥ 1, then either ψ = 0 or ψ(C0 ) ⊂ C0 . (2) If c0 = 0, then ψ(C0 ) ⊂ C1/2 . A detailed exposition of that result is the main goal of Chap. 8. Further work on composition operators on spaces of Dirichlet series can be found in [16–19].

6.4.3 Multipliers of H2 The Banach space H2 is not an algebra (i.e. is not stable under products), but nearly! Indeed, we have the following property, in which d(n) denotes the number of divisors of the integer n:  ∞ −s −s Theorem 6.4.6 Let f (s) = ∞ ∈ H2 and h(s) = n=1 an n , g(s) = n=1 bn n ∞ −s n=1 cn n , where h = f g and c = a ∗ b. Then, we have

162

6 Hardy Spaces of Dirichlet Series ∞  |cn |2

d(n)

n=1

≤ f 22 g 22 .

(6.4.2)

In particular, σa ( f g) ≤ 1/2.  Proof We have cn = i j=n ai b j . Cauchy-Schwarz gives        2 2 2 2 1 |ai | |b j | = d(n) |ai | |b j | . |cn | ≤ 2

i j=n

i j=n

i j=n

Therefore ∞  |cn |2 n=1

d(n)

≤

∞   n=1

 |ai | |b j | 2

2

=

i j=n

 ∞

|ai |

i=1

2

  ∞

 |b j |

2

= f 22 g 22 .

j=1

Let σ > 1/2, and ε > 0 such that ε < 2σ − 1. As is well-known [20, p. 84]: d(n) ≤ Cε n ε . Cauchy-Schwarz once more gives ∞ 

|cn |n −σ ≤

n=1

 ∞  ∞ |cn |2 1/2  n=1

d(n)

n −2σ d(n)

1/2 < ∞.

n=1

 To see how far H2 is from being an algebra, we have the following basic theorem, which gives a link between both spaces H2 and H∞ [1]: Theorem 6.4.7 (Hedenmalm-Lindqvist-Seip) The space of multi plier s of H2 is exactly the space H∞ . In other words, a function f defined on H2 satisfies f g ∈ H2 for all g ∈ H2 if and only if f ∈ H∞ . Moreover f ∞ =

sup g∈H2 , g 2 ≤1

f g 2 .

(6.4.3)

Proof In the proof of that basic theorem, 2 will designate the H2 -norm. We will follow a different route than in [21], closer in a sense to that of [22], and which also uses a method due to E. Saksman, already alluded to in this chapter ([23], see also [1, 4]). This method can be summarized as follows: denote once and  for all by E  ψ(ξ)  = ψ(t)e−itξ dt, ψ, the set of functions ψ ∈ L 1 (R) whose Fourier transform  R −s ∈ H2 and ψ ∈ is compactly supported. Then, given a function f (s) = ∞ n=1 an n  E, the following vertical convolution identity (à la Fejér, with ψ(log n) instead of + ) ) holds: (1 − |n| N ∞  n=1

 an ψ(log n)n −s =

 f (s + it)ψ(t)dt, s ∈ C1/2 . R

(6.4.4)

6.4 The Hilbert Space H2

163

Two lemmas will first be needed. Lemma 6.4.8 Let f ∈ H∞ and g ∈ H2 . Then, f g ∈ H2 and f g 2 ≤ f ∞ g 2 .

(6.4.5)

Indeed, denote f (s) =

∞ 

an n −s , g(s) =

n=1

∞ 

bn n −s , ( f g)(s) =

n=1

∞ 

cn n −s

n=1

N  (N ) −s then let g N (s) = n=1 bn n −s and ( f g N )(s) = ∞ where N is a positive n=1 cn n integer. One applies Theorem 6.2.5 (Carlson’s identity) to the function f g N ∈ H∞ . For a given ε > 0, one then gets: ∞ 

|cn(N ) |2 n −2ε

n=1

≤

f 2∞

1 lim T →∞ T

1 = lim T →∞ T

T | f (ε + it)|2 |g N (ε + it)|2 dt 0

T |g N (ε + it)|2 dt = f 2∞

N 

|bn |2 n −2ε ≤ f 2∞ g 22 .

n=1

0

 2 (N ) 2 2 Hence, letting ε tend to 0, one gets ∞ n=1 |cn | ≤ f ∞ g 2 by Fatou’s lemma. Finally, we observe that, for each fixed n, it holds cn(N ) =



N →∞

ai b j →

i j=n, j≤N



ai b j = cn

i j=n

and we apply Fatou’s lemma again. In the forthcoming lemma, for an integer r ≥ 1, we will denote by Nr the multiplicative semi-group generated by the first r primes p1 , . . . , pr , i.e. the set of positive integers whose prime factors are all ≤ pr . And by Pr the set ofDirichlet polynomials −s satisfying: with spectrum inside Nr , namely those polynomials f (s) = ∞ n=1 cn n cn = 0 ⇒ n ∈ Nr . Lemma 6.4.9 For each s ∈ C0 and each integer r ≥ 1, there exists a constant Cs,r > 0 such that (6.4.6) f ∈ Pr ⇒ | f (s)| ≤ Cs,r f 2 . Proof Write s = σ + it ∈ C0 and f (s) = identity shows that

 n∈Nr

cn n −s ∈ Pr . A truncated Euler

164

6 Hardy Spaces of Dirichlet Series

Cs,r :=



n −2σ =

n∈Nr

r 

(1 − p −2σ )−1 < ∞. j

j=1

The Cauchy-Schwarz inequality now gives | f (s)| ≤



|cn |

2

1/2  

n∈Nr

n

−2σ

1/2 = Cs,r f 2 .

n∈Nr

Let us come back to the proof of Theorem 6.4.7. If f is a multiplier of H2 , denote by M f : H2 → H2 the operator of multiplication by f . We have f ∈ H2 because f = M f (1), and M f is a bounded operator, thanks to the closed graph theorem. We denote its norm by λ(= λ f ). Lemma 6.4.8 tells us that any function f ∈ H∞ is a multiplier satisfying λ ≤ f ∞ . We will prove the converse in three steps. Step 1 (“Power trick”). One has λ = f ∞ if f is a Dirichlet polynomial. Indeed, fix r such that f ∈ Pr and s ∈ C0 . One inductively gets f k 2 ≤ λk for every integer k ≥ 1, since f k = M f ( f k−1 ) and f 0 = 1. Lemma 6.4.9, applicable to f k ∈ Pr (a key point), gives for fixed s ∈ C0 : 1/k  . | f (s)| ≤ Cs,r f 2 ≤ Cs,r λ , then | f (s)| ≤ λ Cs,r k

k

k

Letting k tend to infinity, we get | f (s)| ≤ λ and finally f ∞ ≤ λ. Step 2. ∞  −s −s  Write f (s) = ∞ n=1 an n . Let ψ ∈ E and P(s) = n=1 an ψ(log n)n , a Dirichlet polynomial. Then (6.4.7) M P ≤ M f × ψ 1 . Indeed, for given real t, denote by Tt the vertical translation operator by it defined by Tt g(s) = g(s + it). This operator is an isometry of H2 and moreover  Tt f is again  a multiplier, satisfying MTt f = M f . Indeed, since (Tt f )g = Tt f (T−t g) , we see that (Tt f )g 2 = f (T−t g) 2 ≤ M f T−t g 2 = M f g 2 . Now, Saksman’s   vertical convolution formula (6.4.4) writes as a vector-valued integral (with = R ):  P=

(Tt f ) ψ(t)dt.

 Hence, for g ∈ H2 , Pg = (Tt f g) ψ(t)dt and   Pg 2 ≤ Tt f g 2 |ψ(t)|dt ≤ M f g 2 |ψ(t)|dt = M f g 2 | ψ 1 , which establishes (6.4.7)

6.4 The Hilbert Space H2

165

Step 3. Let (ψ j ) j≥1 ⊂ E be an approximate identity in L 1 , for example a sequence  + |ξ|  . And let P j be the corresponding Dirichlet polynosuch that ψ j (ξ) = 1 − j mial: P j (s) =

∞ 

j (log n)n −s = an ψ

 (Tt f )(s) ψ j (t)dt

n=1

by (6.4.4). The first two steps show that P j ∞ = M P j ≤ λ. Taking a subsequence if needed, one can assume, by Montel’s theorem, that P j tends to some function subsets of C0 , with F ∞ ≤ λ. Let now s = F ∈ H ∞ (C0 ) uniformly  on compact −σ j (log n) → 1 as j → ∞, with 0 ≤ |a |n < ∞ and ψ σ + it ∈ C1/2 . Then, ∞ n=1 n j (log n) ≤ 1, so that ψ P j (s) →

∞ 

an n −s = f (s) as j → ∞,

n=1

showing that f (s) = F(s). Hence, f can be extended to an analytic function, bounded by λ on C0 . And f ∈ D since f ∈ H2 .  Remark Let H p be the Hardy-Dirichlet space coined just afterwards in this chapter. The previous result can be effortlessly generalized to the multipliers M(H p ) of H p , 1 ≤ p ≤ ∞. Namely, we have the following theorem. Theorem 6.4.10 The multipliers of H p can be isometrically identified with H∞ . More precisely, let f be a complex function defined on C1/2 . Then f g ∈ H p for all g ∈ H p ⇔ f ∈ H∞ , and even f ∞ = sup f g p . g p ≤1

Proof The preceding steps of the case p = 2 work. Indeed, we can assume that 2 < p < ∞, since otherwise M(H p ) ⊂ M(H2 ). With the previous notations, one has since f g N ∈ H∞ ⊂ H p :

f g N pp

1 = lim lim ε→0 T →∞ T

T |( f g N )(ε + it)| p dt 0

≤

p f ∞

1 lim lim ε→0 T →∞ T

T p |g N (ε + it)| p dt = f ∞ g N pp . 0

This implies f g p ≤ f ∞ g p . And for the converse, Lemma 6.4.9 works as well since . 2 ≤ . p for p > 2. The final argument with vector-valued integrals is the same. 

166

6 Hardy Spaces of Dirichlet Series

6.5 The Banach Spaces H p 6.5.1 A Basic Identity Let p ≥ 1. We already defined H2 and H∞ . We wish to define more generally a space H p of Dirichlet series, which would be the analogue of the Hardy space H p on the the unit disk. To that effect, we recall the Birkhoff-Oxtoby theorem of Chap. 2 (in which μ denotes the Haar measure of T∞ and e ∈ T∞ the vector (1, 1, . . . , 1, . . .)): Theorem 6.5.1 (Birkhoff-Oxtoby Theorem for the Kronecker flow) Let us denote by (K t ) be the Kronecker flow on T∞ associated with a sequence (x1 , . . . , xd , . . .) of independent real numbers, and let f : T∞ → C be a continuous function. Then we have, as T → ∞: 1 2T 1 T



T f (K t e)dt → −T

f (z)dμ(z);

(6.5.1)

T∞



T f (K t e)dt →

f (z)dμ(z).

(6.5.2)

T∞

0

 −s (here, n −s Now, denote by P the set of Dirichlet polynomials P(s) = ∞ n=1 cn n appears essentially as a free variable). If P ∈ P, let P(z) = γα z α the associated algebraic polynomial, following the Bohr point of view of Chaps. 3 and 4, where we recall that z α = z 1α1 . . . zrαr and γα = cn if n = p1α1 . . . prαr . With those notations, we have the Theorem 6.5.2 For each P ∈ P and each 1 ≤ p < ∞, one has ⎡ 1 P p := lim ⎣ T →∞ 2T

T

⎤ 1p |P(it)| p dt ⎦ = P L p (T∞ ).

−T

In particular, P → P p is a norm on P. Proof Let x j = log p j where p j is the jth prime number, and let (K t ) be the Kronecker flow associated with this independent sequence. Let also f (z) = |P(z)| p , which is continuous and clearly verifies f (K t e) = |P(it)| p . Applying the BirkhoffOxtoby theorem above to f , we at once obtain Theorem 6.5.2, since moreover P p = 0 =⇒ P L p (T∞ ) = 0 =⇒ P = 0 =⇒ P = 0. 

6.5 The Banach Spaces H p

167

6.5.2 Definition of Hardy-Dirichlet Spaces The Dirichlet space H p , p < ∞, will be defined as the completion of Dirichlet polynomials for the norm p above. This concretely means the following: a formal an n −s is in H p if there exists a Cauchy sequence (for the p Dirichlet series ∞ n=1 ( j) ∞ norm) P j (s) = n=1 an n −s of Dirichlet polynomials such that lim an( j) = an for all n = 1, 2, . . .

j→∞

With this definition, the following first two items are easy, since P L p (T∞ ) ≤ P L q (T∞ ) for p ≤ q. Proposition 6.5.3 Let 1 ≤ p ≤ q ≤ ∞. Then: (1) The space H p is a Banach space, isometric to H p (T∞ ). (2) Hq ⊂ H p and f ∈ Hq =⇒ f p ≤ f q . (3) The new definition of H p coincides with the previous one for p = 2. Proof (1) Recall that, by definition, H p (T∞ ) = { f ∈ L p (T∞ );  f (α) = 0 if α ∈ / N(∞) }. ∞ with all coordinates non-negative, i.e. the  The set of all elements α ∈ Z(∞) = T set N(∞) , is called the narrow cone by Helson. The rest is an easy consequence of Theorem 6.5.2 and of a density argument. (2) This is clear from the identity of Theorem 6.5.2. (3) This is clear, since the “new” space H2 is just the completion of the prehilbertian space of finitely supported sequences of complex numbers. 

Remark A nice property of the Banach space H p was established by Aleman et al. [23], under the following form. Theorem 6.5.4 (Aleman–Olsen–Saksman) Let 1 < p < ∞. Then, the  sequence −s (n −s )n≥1 is a Schauder basis of H p . In other terms, if the function f (s) = ∞ n=1 an n p p is in H , the partial sums of that function converge to it in H . Indeed, the authors give two independent proofs, one based on abstract considerations of harmonic analysis on ordered groups, in the spirit of Chap. 1, another one based on a measure preserving change of variable formula which was first used by Fefferman (whether H1 has a Schauder basis, or H p an unconditional basis, seem two open problems). That theorem, and the previous proposition, say nothing of a half-plane where  −s a n ∈ H p would converge pointwise. the Dirichlet series of some element ∞ n n=1

168

6 Hardy Spaces of Dirichlet Series

Since the biggest space is H1 , a nice theorem, due to Helson [24], will give a complete answer, and is a significant extension of Theorem 6.4.6. But the proof requires preliminaries of harmonic analysis which we begin by describing.

6.5.3 A Detour Through Harmonic Analysis 6.5.3.1

L 1 and L 2 -Norms

We begin with a version of Hardy’s inequality, which expresses that the canonical injection of the Hardy space H 1 into the Bergman space B 2 is a contraction. Recall that the usual Hardy inequality is as follows: If f (z) =

∞ 

cn z n ∈ H 1 , then

n=0

∞  |cn | ≤ π f 1 . n+1 n=0

The next inequality, independently due to Helson [24] and Vukotic [25], is in some respects worse, since it involves a 2 -norm instead of a 1 -norm, but in some others better, since we get rid of the constant π > 1. It will lend itself to iterations, which would not be the case of Hardy’s inequality.  n 1 Lemma 6.5.5 Let f (z) = ∞ n=0 cn z ∈ H . Then, we have the contractive Hardy inequality: 1  ∞ |cn |2 2 ≤ f 1 . (6.5.3) n+1 n=0  n Proof Let f (z) = ∞ n=0 cn z . We can assume that f 1 = 1. We use the classical and usual fact according to which f can be written as f = gh with g 2 = h 2 = 1, or cn =

n 

ak bn−k where

k=0 ∞  k=0

|ak |2 =

∞ 

|bk |2 = 1.

k=0

We next use Cauchy-Schwarz without separating the ak and the bn−k . We get ∞

 |cn |2 ≤ |ak |2 |bn−k |2 . n+1 k=0 Summing with respect to n and √ using a Cauchy product in the right-hand-side, we get (6.5.3). And since r ≤ 1/ 2 =⇒ r 2n ≤ 1/(n + 1)∀n ≥ 0, (6.5.3) proves in

6.5 The Banach Spaces H p

169

√ passing the contractivity of the Poisson kernel Pr : H 1 → H 2 for r ≤ 1/ 2, a result first √ proved in [26] in the general case of Pr : H p → H q (contractivity holds iff r ≤ p/q). This notion of “hypercontractivity” had been introduced by Bonami in her thesis (see [27]), for other semi-groups.  As we already mentioned, this lemma can be iterated using a clever induction and a reverse Hölder inequality to give [24]:  α 1 r Lemma 6.5.6 (Helson) Let f (z) = ∞ α∈Nr cα z ∈ H (T ), r ≥ 1. For a multir index α = (α1 , . . . , αr ) ∈ N , set w(α) = (1 + α1 ) · · · (1 + αr ). Then, we have  α∈Nr

|cα |2 w(α)

 21

≤ f 1 .

(6.5.4)

Proof The case r = 1 is the previous lemma. For the induction, we can assume by density that f is a polynomial, we take the case of r − 1 variables for granted, and pass to r variables. We need the following well-known integral form of Minkowski’s inequality. Lemma 6.5.7 Let ρ be positive and measurable on the product (X, d x) × (Y, dy) of two measured spaces. Then: ⎤2 ⎞1/2 ⎤1/2 ⎡ ⎛ ⎡     2 ⎝ ⎣ ρ(x, y)d x ⎦ dy ⎠ ≤ ⎣ ρ (x, y)dy ⎦ d x. Y

X

X

(6.5.5)

Y

Note that, setting ρx (y) = ρ(x, y), (6.5.5) should be read as         ρx d x  ≤ ρx 2 d x.     X

X

2

Now, defining the operators T1 , T2 , . . . , Tr by T j f (z 1 , . . . , zr ) =

 c(α1 , . . . , αr ) α  z 1 1 . . . zrαr , 1 ≤ j ≤ r. α + 1 j j

we have to prove: T1 T2 · · · Tr f 2 ≤ f 1 . We write z j = e2iπt j , 0 ≤ t j ≤ 1. First, Lemma 6.5.5 gives us  [0,1]

⎡ |T1 · · · Tr f (t1 , . . . , tr )|2 dt1 ≤ ⎣



[0,1]

⎤2 |T2 . . . Tr f (t1 , . . . , tr )|dt1 ⎦ .

170

6 Hardy Spaces of Dirichlet Series

Now, we integrate with respect to t2 , . . . , tr and use Lemma 6.5.7 to obtain ⎡



⎣

T1 T2 · · · Tr f 22 ≤ ⎛ ⎜ ≤⎝

⎡





[0,1]

⎤2 |T2 . . . Tr f (t1 , . . . , tr )|dt1 ⎦ dt2 · · · dtr

[0,1]

[0,1]r −1

⎢ ⎣



⎤1/2

⎥ |T2 . . . Tr f (t1 , . . . , tr )|2 dt2 · · · dtr ⎦

⎞2 ⎟ dt1 ⎠ .

[0,1]r −1

Finally, we use the induction hypothesis and get ⎛



⎜ T1 T2 · · · Tr f 22 ≤ ⎝

⎡ ⎢ ⎣

[0,1]

⎤



⎞2

⎥ ⎟ | f (t1 , . . . , tr )|dt2 · · · dtr ⎦ dt1 ⎠ = f 21

[0,1]r −1

ending the proof of Lemma 6.5.6.  Two nice applications of Lemma 6.5.6 are a “dimension-free” inequality and the theorem of Helson previously mentioned. Proposition 6.5.8 Let P(r, d) be the set of homogeneous polynomials of degree d in r variables, namely the set of P(z) =



cα z α with α = (α1 , . . . , αr ) and |α| = α1 + · · · + αr = d.

α

If P ∈ P(r, d), it satisfies: P 2 ≤ 2d/2 P 1 .

(6.5.6)

Proof For α = (α1 , . . . αr ) and |α| = d, note that w(α) = (1 + α1 ) · · · (1 + αr ) ≤ 2α1 · · · 2αr = 2d . We then have from Lemma 6.5.6: P 2 =



|cα |2

"1/2

1/2   |cα |2 ≤ 2d ≤ 2d/2 P 1 . w(α)

Theorem 6.5.9 (Helson) Let f (s) =  ∞ n=1

∞

|cn |2 d(n)



n=1 cn n

−s

∈ H1 . Then, we have

1/2 ≤ f 1

6.5 The Banach Spaces H p

171

where d(n) denotes the number of divisors of n. In particular, σa ( f ) ≤ 1/2.   Proof Let f (s) = n cn n −s be a Dirichlet polynomial, and let  f (z) = α γα z α α1 be the associated algebraic polynomial (Bohr’s point of view). If n = p1 · · · prαr , we have d(n) = (1 + α1 ) · · · (1 + αr ) =: w(α). Therefore, using Lemma 6.5.6 and Theorem 6.5.2, we get   |cn |2 1/2 n

d(n)

=

  |γα |2 1/2 ≤  f 1 = f 1 w(α) α

and by density the inequality holds for any f ∈ H1 . And it implies that σa ( f ) ≤ 1/2  since d(n) ≤ Cε n ε for each ε > 0.

6.5.4 Helson Forms In this section, we will see that Theorem 6.5.9 has other applications, besides the ones to be found in [25]. In his last two papers [24, 28], Helson considered bilinear forms (equivalently bounded operators) on the Hilbert space H = 2 (N∗ ) associated with a complex sequence c = (cn )n≥1 , formally defined by H (c)(x, y) =



c jk x j yk .

(6.5.7)

j,k

We thus consider matrices (a j,k ) where the ( j, k)th entry only depends on the product jk, similar to Hankel matrices (the entries only depend on the sum of indices). It is known (Nehari’s theorem) that the continuous Hankel matrices are those with a ( j + k) where ϕ ∈ L ∞ (T). Helson examined the analogue symbol, namely a j,k = ϕ for what we now call Helson matrices. See also [29] for more on this. To understand properly this analogy, recall that, with Bohr’s notations, we have a semi-group isomomorphism h : N(∞) → N∗ defined for α = (α1 , . . . , αr , 0, . . . , 0, . . .) ∈ N(∞) by h(α) = p1α1 · · · prαr , with h(α + β) = h(α)h(β). ∞ with the action α(z) =  We have N(∞) ⊂ T We will once more consider the weight

w(α) =

r 

#

α

j

z j j for α = (α j ) and z = (z j ).

(1 + α j ) = d[h(α)].

i=1

Let (a j,k ) = (c jk ), j, k ≥ 1 be an infinite matrix, and let γα = ch(α) . We will say that this matrix has a symbol if there exists ϕ ∈ L ∞ (T∞ ) such that

172

6 Hardy Spaces of Dirichlet Series

γα = ϕ (α) for all α ∈ N(∞) .

(6.5.8)

In the two papers [24, 28], Helson proved: Theorem 6.5.10 (Helson) Let (a j,k ) = (c jk ), j, k ≥ 1 be an infinite matrix. (i) If this matrix has a symbol, then it defines a continuous operator H (c) on 2 , with the additional property that H (c) ≤ ϕ ∞ . (ii) Conversely, if we assume this matrix to be Hilbert-Schmidt, i.e. S 2 :=



∞ 

|a j,k |2 =

d(n)|cn |2 =

w(α)|γα |2 < ∞,

α

n=1

j,k



then this matrix has a symbol. Proof (i) Using the isomorphism h, it amounts to the same (back to Hankel forms in several variables) to prove the boundedness of the bilinear form 

B(x, y) =

γα+β xα yβ

α,β

on the space of finitely supported vectors of K = 2 (N(∞)  ). If μ is the ∞ , if x = (x ), y = (y ) and if P(z) = Haar measure of T α β α x α α(−z) and  Q(z) = β xβ β(−z), we clearly have B(x, y) =



 ϕ(z)α(−z)β(−z)dμ(z)

xα yβ

α,β



T∞

ϕ(z)P(z)Q(z)dμ(z),

= T∞

whence |B(x, y)| ≤ ϕ ∞ P Q 1 ≤ ϕ ∞ P 2 Q 2 = ϕ ∞ x 2 y 2 , giving the result. (ii) Let L be the linear form on H 1 (T∞ ) given by L( f ) =



γα  f (α).

α∈N(∞)

This linear form is well-defined. Indeed, by Cauchy-Schwarz, the assumption f√(α)) is in 1 since of the theorem and Lemma 6.5.6, the family (γα  √ it is the 2  pointwise product of the two  -sequences f (α)/ w(α) and γα w(α), with moreover

6.5 The Banach Spaces H p

173

|L( f )| ≤



|γα  f (α)| ≤ S f 1 ,

α

  2 2 (using the relation α |γα |2 w(α) = ∞ n=1 |cn | d(n) = S ). By the Hahn-Banach theorem, L can be extended to a continuous linear form (still denoted by L) on L 1 (T∞ ). Therefore, there exists ϕ ∈ L ∞ (T∞ ) such that  L( f ) = T∞ f (z)ϕ(−z)dμ(z) for all f ∈ H 1 (T∞ ). Specializing to the mono(α). mial f (z) = z α , we get L( f ) = γα = ϕ  In a second, moving, paper, which was his ultimate one, Helson returned to that question ([28]) and proved in particular the following (indeed, the first statement of Theorem 6.5.10 is given without proof in this second paper): Theorem 6.5.11 (Helson) There exist matrices (a j,k ) = (c jk ), j, k ≥ 1 which define a continuous operator on 2 , but are not Hilbert-Schmidt. Proof If this were not the case, we would have two comparable Banach norms ∞ (operator norm) and H S (Hilbert-Schmidt norm) on the set of Helson forms, with obviously ∞ ≤ H S , and by the open-mapping theorem there would exist a numerical constant λ such that H (c) H S ≤ λ H (c) ∞ for all forms H (c).

(6.5.9)

To get a contradiction, consider the form with cn = 1 for 1 ≤ n ≤ N , and cn = 0 for n > N . Using a classical estimation in Number Theory ([20, p. 38]), we first get H (c) H S =

 N

1/2 d(n)

∼



N log N .

(6.5.10)

n=1

To get an upper bound on H (c) ∞ , one uses the Schur test: if (a j,k ) j,k≥1 is a symmetric matrix with non-negative entries, recall that this Schur test [30, Problem 45], under a simplified form which is sufficient here, claims. Proposition 6.5.12 Let (a j,k ) j,k≥1 be a symmetric matrix of non-negative numbers. Suppose that there exists a positive sequence p = ( pk )k≥1 , and a positive number M > 0 such that ∞  a j,k pk ≤ M p j for all j. k=1

Then, this matrix is continuous on 2 and the associated operator A  (namely Ax = y, y j = k≥1 a j,k xk ) satisfies: A ≤ M.

174

6 Hardy Spaces of Dirichlet Series

√ Using that test with the symmetric matrix A = H (c) and the weight pk = 1/ k, we see here that N /j  1 √ dx a j,k pk = c jk pk = √ ≤ √ = 2 N pj. x k k≥1 k≥1 jk≤N 



0

Therefore, Proposition 6.5.12 gives us √ H (c) ∞ ≤ 2 N .

(6.5.11)

Now, for large N , the relations (6.5.10) and (6.5.11) are incompatible with (6.5.9), which proves Helson’s theorem by contradiction.  Helson asked if the existence of a symbol still holds if we only assume that the matrix (c jk ) defines a continuous linear operator on 2 . According to his guess (see Subsect. 6.5.5 to follow), the answer was negative. Helson was right, as was shown fairly recently by Ortega-Cerdà and Seip [31]: Theorem 6.5.13 (Ortega-Cerdà–Seip) There exist continuous Helson forms without symbols. Proof If the statement is wrong, by the open mapping theorem, there exists some constant C such that, for every bounded Helson bilinear form on 2 × 2 , B (x, y) =



c jk x j yk ,

j,k≥1

one has ch(α) = ϕ  (α), α ∈ N(∞) , for some ϕ ∈ L ∞ (T∞ ) verifying that ϕ ∞ ≤ B . C Now, consider a large even integer d = 2N and set % $ N  I = n ∈ N; n = q j and q j = p2 j−1 or q j = p2 j , j=1

as well as cn = 1 if n ∈ I and cn = 0 if n ∈ / I. One understands better the motivation by considering the generating function of the sequence (cn ): f (z) =

N  

   −1 cn z h (n) = z 2 j−1 + z 2 j = (α)z α

j=1

  where z = z j ∈ T∞ . We will show that

n

α

6.5 The Banach Spaces H p

175

 π "N ϕ  (α) = ch(α) for all α ∈ N(∞) =⇒ ϕ ∞ ≥ 2 √ " N B ≤ 2 .

(6.5.12) (6.5.13)

√ Since π/2 > 2, the inequality ϕ ∞ ≤ C B will be contradicted for large N . To prove (6.5.12), we first observe that   z 2 j−1 + z 2 j  = 1 1 2π

π −ß

  1 + eit  dt = 4 , so that f 1 = π

 N 4 . π

Next, identifying x = (cn ) and its generating function f , we have, as was seen in the proof of Theorem 6.5.10: 

cj =

j



c2j = |B(x, 1)| = |B ( f, 1)| ≤ ϕ ∞ f × 1 1

j

that is 2 N ≤ ϕ ∞ (4/π) N , giving the result. To prove (6.5.13), we use the Schur test with the weight ( p j ) defined by: p j = 2−ω( j)/2 if j divides some member of I and p j = 0 otherwise. # Now, fix j such that the c jk ’s are not all zero. Then, j must be of the form j = i∈U qi∗ where qi∗ = p2i−1 or p2i , with U ⊂ {1, . . . , N } and |U | = ω( j). We then have 

c jk pk =

k



2−ω(k)/2 .

jk∈I

Set # V = {1, . . . , N }\U . Any index k of the summand must be of the form k = i∈V qi with qi = p2i−1 or qi = p2i and ω(k) = N − ω( j), so that 

c jk pk =

k



2−(N −ω( j))/2 = 2|B| 2−(N −ω( j))/2

jk∈I

= 2 N −ω( j) 2−(N −ω( j))/2 = 2 N /2 p j , proving (6.5.13) and thereby Theorem 6.5.13.



Remark It is well-known (and we used it!) that any f ∈ H 1 (D) can be written as 1/2 remarkable, fairly f = gh, with g 2 = h 2 = f 1 . Similarly, and those are  recent, results, any function f ∈ H 1 (Dd ) can be written as f = k gk h k , with 

gk 2 h k 2 ≤ Cd f 1 .

k

See [32] for the case d = 2 and [33] for the case d > 2. In other terms, H 1 (Dd ) is the projective tensor product of H 2 (Dd ) with itself. What was proved above is that

176

6 Hardy Spaces of Dirichlet Series

the constant Cd is not bounded in d, and indeed increases exponentially in d. As a consequence, the Hardy-Dirichlet space H1 is not the projective tensor product of H2 with itself. This makes Theorem 6.5.9 all the more remarkable.

6.5.5 Harper’s Breakthrough and Two Consequences N Let D N (s) = n=1 n −s be the N th-partial sum of the Riemann zeta function. Its restriction D N (−it) to the boundary ∂C0 of C0 is called the multiplicative Dirichlet kernel of index N . In Harper’s language [34], for given t, the map n → n it is a completely multiplicative random character denoted f (n), seen in the Bohr group of R. We can also think of f (n) as f (n) = z α with z = (z j ) if n = p α = h(α), this time seen on the group T∞ ; both groups being equipped with their Haar measure. Recently, Harper [34] proved the following deep result. N  f (n) = h(α)≤N z α the mulTheorem 6.5.14 Let f be as above and D N = n=1 tiplicative Dirichlet kernel of index N . Then, for every 0 < p < 2, it holds (with constants a > 0 and b > 0 independent of p): √ b N

√ a N

'1/2 ≤ D N p ≤ & '1/2 · & √ √ 1 + (1 − 2p ) log log N 1 + (1 − 2p ) log log N In particular:

√ D N p = o( N ).

The proof is very difficult, and will not be given here. But we will see how it provides a positive answer to a conjecture of Helson, and to an important question about the so-called “local embedding”. Helson’s belief was that some bounded Hankel forms have no symbols. He made to this effect the following observation. Theorem 6.5.15 If every Hankel form has a symbol, then √ D N 1 ≥ δ N where δ is a positive constant.

√ As a consequence, Helson hoped that D N 1 = o( N ), and this hope became popular in the domain under the name “Helson’s conjecture”. Later on, various authors, including Helson himself and Harper, obtained sharper and sharper lower estimates which rather went in the direction that perhaps this conjecture was wrong [35–37]. The statements were respectively:

6.5 The Banach Spaces H p √ N 1/4 . (log √ N) N with some (log N√ )δ N . (log log N )3+o(1)

• D N 1  • D N 1  • D N 1 

177

0 < δ < 1/4.

As Theorem 6.5.14 shows, the conjecture is true, be it only by a very tiny (log log)1/4 factor!! And even though Helson’s belief mentioned above was shown to be the truth first by Ortega-Cerdà and Seip [31] independently of the conjecture, it is instructive to see that the positive answer to this belief also follows from Harper’s Theorem 6.5.14. A word of comment here: Helson’s argument can be seen as a happy “accident”, because it transforms the original problem about Nehari’s theorem (a purely multiplicative and operator theoretic problem) into a hard Number Theoretical problem. The Ortega-Cerdà and Seip approach is probably the “natural” one. Proof The proof of Theorem 6.5.15 is quite close to that of the above Proposition 6.5.11. Under our assumption, the open mapping theorem provides us with a constant λ such that, for every bounded Hankel form H (c), there exists ϕ ∈ L ∞ (T∞ ) such that ϕ (α) = γα for each α ≥ 0 and ϕ ∞ ≤ λ H (c) .

(6.5.14)

Now, we test (6.5.14) on √the Hankel form H (c) of Proposition 6.5.11, for which we know that H (c) ≤ 2 N . This provides us with ϕ ∈ L ∞ (T∞ ) satisfying √ ϕ (α) = 1 if h(α) ≤ N , = 0 for h(α) > N , and ϕ ∞ ≤ 2λ N . We hence obtain, as in the proof of Theorem 6.5.10, that N=



√ ϕ (α) D N (α) ≤ ϕ ∞ D N 1 ≤ 2λ N D N 1 ,

α∈Z(∞)

implying D N 1 ≥

√

N · 2λ



The second consequence of Theorem 6.5.14 concerns the local embedding question: does there exist, for each p > 0, a constant C p such that 1 p

| f (1/2 + it)| p dt ≤ C p f H p for each f ∈ H p ?

(6.5.15)

−1

As we saw in Chap. 1, using the Montgomery-Vaughan generalized Hilbert’s inequality for example, (6.5.15) holds for p = 2 and this will play an essential role in Chap. 8 in the study of composition operators. But the situation is different for 0 < p < 2.

178

6 Hardy Spaces of Dirichlet Series

Proposition 6.5.16 The local embedding (6.5.15) fails for 0 < p < 2. Proof Recall the Euler-MacLaurin summation formula at order 1, in which g denotes a complex C 1 -function on [1, ∞[, N a positive integer and {x} the fractional part of x: N N  g(n) = g(x)d x + {x}g  (x)d x. 1 T .−σThen, for f (s) = n=1 an n ∈ H , we have σ > σa ( f ) and < ∞. We thus have a well-defined map n=1 |an |n L : H∞ → 1 ,

  L( f ) = an n −σ .

This map has closed graph by Theorem 6.1.1 and is therefore continuous. If Cσ = L , specializing, we get N 

|an |n

−σ

   N  −s   ≤ Cσ  an n 

n=1

which implies N 

∞

n=1

   N  −s  |an | ≤ Cσ N σ  a n n  

n=1

∞

n=1

and σ ∈ E. This gives ]T, ∞[⊂ Eand inf E ≤ T . n −s ∗ 2. Let σ ∈ E and f ∈ H∞ , f (s) = ∞ n=1 an n . Set An = j=1 |a j | and fix α > σ. An Abel’s summation by parts shows that N 

|an |n −α =

n=1

N −1 

A∗n [n −α − (n + 1)−α ] + A∗N N −α with

n=1

   n  −s  A∗n ≤ Cσ n σ  a j j   j=1

∞

≤ CCσ n σ log n f ∞

where we used Theorem 6.2.2 and the fact that σ ∈ E. It follows that |an |n −α < ∞, so that successively

∞ n=1

σa ( f ) ≤ α, σa ( f ) ≤ σ, σa ( f ) ≤ inf E, T ≤ inf E, T = inf E, 

ending the proof.

As we already noticed in Chap. 5, the new definition of T poses the question of the best constant S( N ) such that, for all scalars a1 , . . . , a N : N 

  N   it   |an | ≤ S ( N )  an n  ,

n=1

the so-called Sidon constant of the set

n=1

∞

(6.6.2)

180

6 Hardy Spaces of Dirichlet Series

 N = {log 1, log 2, . . . , log N } ⊂ R. Lower bounds were already examined in Chap. 5. The rate of increase of S( N ) has been the object of intense study over the last ten years. The reader will find a very complete and up-to-date account till 2006 in [21]. One quite recent result is due to Defant, Frerick, Ortega-Cerdà, Ounaies, and Seip [39]. This is the object of the rest of this section, and its presentation will require some effort.

6.6.2 Symmetric Multilinear Forms As Bohnenblust and Hille demonstrated, the Bohr point of view inevitably leads to the consideration of multilinear forms. This is what we consider in this seemingly unrelated subsection, which will end up with a quite sharp version of the BohnenblustHille answer. n is the space Cn We will use the following notations: I = {1, 2, . . . , n}, X = l∞ equipped with the sup-norm, the open unit ball of which is the polydisk Dn , and B : X d → C is a symmetric d-linear form with norm B = We use the notation

sup z (1) ≤1,..., z (d) ≤1

|B(z (1) , . . . , z (d) )|.

B(z (1) , . . . , z (d) ) =

 →

i

with

b→ z → i

i

∈I d

→

(1) (d) i = (i 1 , . . . , i d ) , b→i = b (i 1 , . . . , i d ) and z →i = z i1 . . . z id .

To B is naturally associated a homogeneous polynomial P of n variables with degree d, defined by P (z) = B(z, . . . , z) and one has to understand the relation between B and P. Write   → cα z α = b→ z → with z → = z i1 . . . z id if i = (i 1 , . . . , i d ) . P (z) = |α|=d

→

i

i

i

i

We claim that the connection between the cα ’s and the b→’s is as follows: given →

i

i , we set  → → → → " αs ( i ) = #{k ∈ [1, d] ; i k = s} and α( i ) = α1 ( i ), . . . , αn ( i )

6.6 A Sharp Sidon Constant

181

→

→

and we denote by | i | the “valency” of the multi-index i , namely: $ % → → → → | i | = # j ; α( j ) = α( i ) . Then, we have the two simple facts: →

|i |=

d!

and b→ =

→

α( i )! B ≤

1

cα(→) .

(6.6.3)

dd P ∞ ≤ ed P ∞ . d!

(6.6.4)

i

→

|i |

i

The first fact is easy combinatorics, the second a classical polarisation formula, which can be proved using a Rademacher sequence (εn ): (1)

B(z , . . . , z

(d)

   d 1 (i) . ) = E ε1 . . . εd P εi z d! i=1

Then, (6.6.4) follows immediately. We will make use of the following inequality, due to Blei [40]. Theorem 6.6.2 (Blei) Let (a→)→∈I d be an array of complex numbers, indexed by the i i set I d . Then, one has for pd = 2d/(d + 1): 

1/ pd |a | →

i

→

i

≤

pd

n 

"1/d Ak

,

(6.6.5)

k=1

∈I d

where Ak =

 j∈I



1/2 |a→|2 i

→

.

i ∈I d , i k = j

If the array is symmetric, all the Ak ’s are equal, and we have 

1/ pd |a | →

→

i

i

pd

≤ Ak .

(6.6.6)

∈I d

The initial proof of (6.6.5), written in the more general context of fractional cartesian products, is elementary, but tricky, and not so easy to follow. In the simpler context of truly cartesian products, which is sufficient for our purposes, here is a very clear proof kindly indicated to us by Seip [41]. This proof needs certain flexible versions of the Minkowski and Hölder inequalities which we begin by recalling.

182

6 Hardy Spaces of Dirichlet Series

Lemma 6.6.3 (Minkowski’s inequality) Let (ai, j )1≤i≤M be a rectangular (M × N ) 1≤ j≤N

matrix of non-negative numbers and 1 < r < ∞. Then

  1/r   1/r ( ai, j )r ≤ ai,r j . i

j

j

i

Indeed, let us equip R M with the norm (u i ) r = ( to apply the triangle inequality in that space, namely

 i

|u i |r )1/r . Now, it is enough

      X j X j r  ≤  r

j

j

to the vectors X j = (a1, j , . . . , a M, j ), 1 ≤ j ≤ N .



Lemma 6.6.4 (Hölder’s inequality) Let E be a finite set, let f 1 , . . . , f d : E → R+ be d d non-negative functions, and q1 , . . . , qd be d positive numbers such that k=1 1/qk = 1. Then, omitting the variable j in the f k ’s, we have 

f1 · · · fd ≤ (

j∈E



q

f 1 1 )1/q1 · · · (

j∈E



q

f d d )1/qd

j∈E

  q q Indeed, we may assume that j∈E f 1 1 = · · · = j∈E f d d = 1. Then, the concavity of the logarithm function gives, for fixed j ∈ E: q

q

f1 · · · fd ≤

f d f1 1 + ··· + d . q1 qd

 By summation, we get the desired inequality, since dk=1 1/qk = 1. Note that we choose once and for all to suppress the dependence in j in the sum because it simplifies the notation considerably.  Now, to the proof of Theorem 6.6.2. The reader who wants to understand that proof is “condemned” to study the case d = 2, with which we begin. We may assume the a→ to be non-negative. We write a for a(i 1 , i 2 ) and we set i

S=



a 4/3 =

i 1 ,i 2



f 1 f 2 with

f 1 = f 2 = a 2/3 .

i 1 ,i 2

Recall that A1 =

  i1

i2

1/2 a2

and A2 =

  i2

i1

1/2 a2

.

6.6 A Sharp Sidon Constant

183

We apply Hölder’s inequality with respect to i 1 and with the exponents q1 = 3/2 and q2 = 3 to get 

a 4/3 ≤

 2/3  1/3 a a2 =: g1 g2 .

i1

i1

i1

We now sum with respect to i 2 and apply Holder’s inequality to the functions g1 , g2 and the exponents q1 = 3 and q2 = 3/2 to get S≤

&  2 '1/3 &  1/2 '2/3 a a2 . i2

i1

i2

i1 2/3

2/3

With our notations, the second square bracket is A2 and the first one is ≤ A1 by Minkowski’s inequality. We thus get 

S 3/4 ≤

A1 A2

which is Blei’s inequality for d = 2. In the general case, we still assume a = a→ ≥ 0 i and we start by applying Hölder’s inequality with respect to the first coordinate i 1 of →

i = (i 1 , . . . , i d ), with the exponents q1 =

which verify well

d +1 , q2 = · · · = qd = d + 1 2

d  1 2 d −1 = + = 1, q d + 1 d +1 k=1 k

and for the functions f 1 (i 1 ) = · · · = f d (i 1 ) = a 2/d+1 . We obtain  →

i

2d

a d+1 ≤

2   d+1  d−1 ( a) ( a 2 ) d+1 .

i 2 ,...,i d

i1

i1

We next choose f 1 (i 2 ) = · · · = f d−1 (i 2 ) = (



1

a 2 ) d+1 ,

f d (i 2 ) = (

id

and we apply Hölder’s inequality with j = i 2 , which gives

 i1

2

a) d+1

184

6 Hardy Spaces of Dirichlet Series



a

2d d+1

→

1 2  &  2 ' d+1 &   1 ' d+1 2 ≤ a a 2 i ,...,i i i i i 3

i

×

d

2



a2

1

 d−2 d+1

2

1

.

i 1 ,i 2

We continue in this fashion so that in the l-th step we choose f 1 (il ) =

 

a

2

1  d+1

.

i 1 ,...,il−1

After d steps, we get



2d

a d+1 ≤ b1 · · · bd

→

i

where b1 =

1 1  21 2 ' d+1 &   2 ' d+1 &     a , bl = a2

i 2 ,...,i d

il+1 ,...,i d

i1

for 1 < l < d, and bd =

&   id

a

2

i 1 ,...,il−1

il

2  21 ' d+1

.

i 1 ,...,i d−1

We finish the proof by applying Holder’s inequality to b1 , . . . , bd , with the exponents d +1 q1 = · · · = qd−1 = d + 1 and qd = 2  and finally Minkowski’s inequality to each of the factors b1 , . . . , bd−1 . Note that (6.6.6) is an immediate consequence of (6.6.5) since then all the Ak ’s are equal. We will rely on (6.6.6) to prove the following, weighted form, of Blei’s result, which is in some sense the really new argument [39]. Theorem 6.6.5 Let (b→ )→∈I d be a symmetric array of complex numbers, indexed by i

i

the set I d , B and P the corresponding symmetric d-linear form and homogeneous polynomial. Then, one has  p1  ( → d ( | i ||b→ |) pd ≤ 2d B ≤ (2e)d P ∞ , →

i ∈I d

with pd = 2d/(d + 1).

i

(6.6.7)

6.6 A Sharp Sidon Constant

185

→

Proof If j = ( j1 , . . . , jd−1 ) ∈ I d−1 and 1 ≤ i ≤ n, we will denote by →

j i ∈ I d the multi-index ( j1 , . . . , jd−1 , i).

Let ( E be the LHS of (6.6.7): we apply the symmetric version of Blei’s result with a→ = i

→

| i |b→ to get with those notations i

E ≤ F :=

 

→

| j i b → |

j

 21

ji

→

i∈I

2

.

∈I d−1

→

→

Now, we compare the valencies of j and j i and first claim that (recall that α! = α1 ! . . . αn ! for α = (α1 , . . . , αn )): →

→

α( j i)! ≥ α( j )!

(6.6.8)

Indeed, one has for each s ∈ I →

→

→

αs ( j i) = #{k ∈ [1, d]; ( j i)(k) = s} ≥ #{k ∈ [1, d − 1]; j (k) = s} →

→

→

= αs ( j ), so that αs ( j i)! ≥ αs ( j )! and by product we get (6.6.8). As a consequence, using (6.6.3) twice, →

| j i| =

d! →

α( j i)!

≤

d! →

α( j )!

=d

(d − 1)! →

α( j )!

→

= d| j |,

which gives us since E ≤ F:  2 n √   → de f √  2 → | j b | = d Fi . E≤ d 1

i∈I

ji

→

j

(6.6.9)

i=1

∈I d−1

√ The parameter d√ reappears under a softened form d, whereas it appeared in the more violent form d! in the previous versions of the Bohnenblust-Hille theorem [12]. Now, each term in the RHS of (6.6.9) can be interpreted as a L 2 -norm, namely that of the polynomial (denoting by ei the ith vector of the canonical basis of Cn ): Q i (u) = B(u, . . . , u, ei ) =

 →

j ∈I d−1

b→ u → = ji

j

 |α| = d−1

cα u α

186

6 Hardy Spaces of Dirichlet Series

with, using (6.6.3), c

→

→

α( j )

= | j |b → . ji

→

→

Indeed, Parseval’s formula gives us, since each j ∈ I d−1 is repeated | j | times: Fi2 =



→

| j |(|b → |)2 = ji

→

 1 →

j

j

→

| j|

→

(| j b → |)2 = ji



|cα |2 = Q i 22 .

|α|=d−1

Here, we use (6.5.6) of the previous section to get n 

Fi =

i=1

n 

Q i 2 ≤ 2(d−1)/2

i=1

n 

Q i 1 .

i=1

But we have by linearization, m denoting the Haar measure of Tn :  Q i 1 =

 |B(u, . . . , u, ei )|dm(u) =

Tn

B(u, . . . , u, wi (u)ei )dm(u), Tn

where wi is a unimodular, mesurable function. By adding and noting that wi (u)ei l∞n = 1 for all u, we get n 

 Q i 1 =

i=1

B(u, . . . , u,

n 

n i=1

wi (u)ei )dm(u) ≤ B ,

i=1

Tn

and finally, using (6.6.4), E≤

n √  √ d Fi ≤ d2(d−1)/2 B ≤ 2d B ≤ (2e)d P ∞ , i=1



and this ends the proof of Theorem 6.6.5.

6.6.2.1

Estimation of a p-Sidon Constant

 the dual of a compact Recall that the Sidon constant S() of a finite set () ⊂ G, abelian group G, is the smallest constant such that  λ∈

     |aλ | ≤ S() aλ λ  λ∈

∞

for all aλ ∈ C.

6.6 A Sharp Sidon Constant

187

We have S() ≤ ||1/2 by Parseval’s identity. For 1 ≤ p ≤ 2, it will be more useful here to introduce the p-Sidon constant S p () defined as the smallest constant such that     1p   p  |aλ | ≤ S p () a λ λ (6.6.10)  for all aλ ∈ C. λ∈

λ∈

∞

We have S1 () = S() and S2 () = 1. Here, we will consider the case  = Zn ,  = (n, d), with G = Tn , G (n, d) = {α = (α1 , . . . , αn ) ∈ Nn ; |α| = α1 + · · · + αn = d}. In other terms (n, d) is the Fourier spectrum of the set of homogeneous polynomials of n variables with degree d, already seen in Proposition 6.5.8. Observe that those homogeneous polynomials also naturally appear in Bohr’s point of view, which gives an expansion in a series of such polynomials. In our situation, it is the constant S p () which will be well controlled for specific values of p. We will set once and for all (see the previous subsection): pd =

2d , 1 ≤ pd < 2. d +1

(6.6.11)

The main result of this subsection is the following inequality, called the hypercontractive polynomial Bohnenblust-Hille inequality because the implied constant has at most geometric growth with respect to the degree d, and which is dimension-free in that the number n of variables does not appear. This inequality has many applications, in particular to the estimation of the Bohr radius for the polydisk in high  dimension. Namely the biggest r = rn > 0 such that all polynomials α aα z α in n variables satisfy      |aα z α | ≤ sup  |aα z α . sup n n z∈r D

z∈D

α

α

As a basic consequence of the forthcoming Theorem 6.6.6, one finds the optimal estimate [39] ) log n rn ≈ . n This (final) estimate improves on previous results of Bayart-Matheron and DefantFrerick, who independently had found * rn ≥ δ

log n . n log2 n

Theorem 6.6.6 (Defant–Frerick–Ortega–Cerdà–Ounaies–Seip) There exists a numerical constant C > 1 such that

188

6 Hardy Spaces of Dirichlet Series

S pd [ (n, d)] ≤ C d .  Proof Let P(z) = α cα z α a homogeneous polynomial of n variables with degree d. By polarization, we can write P(z) = B(z, . . . , z) where B is a symmetric, dlinear form with coefficients, say, b→ . Let us set p = pd = 2d/(d + 1) and observe i

that p−1 ≤ 1/2 since p < 2. Now, use (6.6.3), (6.6.4) and the basic weighted estimate p (6.6.7). We obtain  

|cα |

p

 1p

=



(| i b |) i →

p

1

 1p

→

|i |  1p    1p  → p−1 → 1 = (| i | p |b→|) p ≤ (| i | 2 |b→|) p i i →

α∈(n,d)

→

i ∈I d

→

→

i ∈I d

i ∈I d

≤ 2 B ≤ (2e) P ∞ . d

d

This ends the proof, with C = 2e.



Remark In [39], the reader will find the sharper estimate 

1 S pd [(n, d)] ≤ 1 + d −1

d−1

√ √ d−1 d( 2) .

6.6.3 The Claimed Sharp Upper Bound The aim of this subsection is to prove the following sharp upper bound for  S( N ), the Sidon constant of {log 1, . . . , log N }. Set log2 = loglog and λ(x) = log x log2 x. Recall that, in Chap. 5, we found the lower bound √ 1 S( N ) ≥ a N exp[−(b0 + o(1))λ(N )] with b0 = √ . 2 We will prove here that the constant b0 is best possible (see [12, 39, 42]. Theorem 6.6.7 We have for S( N ) the upper bound √ S( N ) ≥ a N exp[−(b0 + o(1))λ(N )]

(6.6.12)

√ where b0 = 1/ 2. Proof To use the current notation in analytic number theory, we will set N = x. We refer to Chap. 5 for the three notations P + (n), S(x, y) and (x, y). Let P − (n) be the least prime divisor of a positive integer n, agreeing that P − (1) = 1, and (n)

6.6 A Sharp Sidon Constant

189

the number of prime divisors of n counted with their multiplicity (whereas ω(n) is the number of distinct prime divisors of n). We consider the sets , + T (x, y) := n ≤ x; P − (n) > y , Tk (x, y) := {n ∈ T (x, y) ;  (n) = k} where 3 ≤ y ≤ x is large. In particular, we assume that y ≥ 4C 2 , where C is the constant appearing in Theorem 6.6.6. We also set Nk (x, y) := #{n ∈ T (x, y); (n) ≥ k} =



#{T j (x, y)}.

j≥k

As in [42], we will make use of the following lemma, due to Balazard [43]. See also [12], Lemma 4.2, which gives a quite elementary proof if one does not care about the constant c, whose best value plays no role in the sequel. Lemma 6.6.8 There exists c > 0 such that, for x ≥ y ≥ 3 and k ≥ 1, one has the uniform estimate in k: Nk (x, y) 

x y log2 x+cy x e  k ecy log2 x . k y y

(6.6.13)

 Now, let P(s) = n≤x an n −s be a Dirichlet polynomial, with P ∞ = 1. Let 2 ≤ y ≤ x to be adjusted, and denote by r = π(x) and s = π(y) respectively the indices (note that s ≤ r ) such that pr ≤ x < pr +1 and ps ≤ y < ps+1 .  According to Bohr’s point of view, let f (z) = n≤x an z α(n) the associated polynomial in r variables, which verifies f ∞ = P ∞ = 1. Each integer n ≤ x can be written in a unique way as n = ml with P + (m) ≤ y and P − (l) > y. Indeed, if n = p1α1 (n) · · · prαr (n) , we can solely take α

s+1 m = p1α1 (n) · · · psαs (n) and l = ps+1

(n)

· · · prαr (n) .

We set z = (z 1 , . . . , zr ) =: (u, v) with u = (z 1 , . . . , z s ) and v = {z s+1 , . . . , zr ). Accordingly, we write

190

6 Hardy Spaces of Dirichlet Series

f (z) =

 ml≤x

u α(m)





aml u α(l)



l∈T (x/m,y)

m∈S(x,y)



=:



aml u α(m) v α(l) = u α(m) f m (v).

m∈S(x,y)

We observe that f m ∞ ≤ f ∞ = 1.

(6.6.14)

Indeed, f m is a Fourier coefficient of f : f m (v) =

1 (2π)s



f (eiθ1 , . . . , eiθs , v)e−i(α1 (m)θ1 +···+αs (m)θs ) dθ1 . . . dθs .

Ts

 Let also f m,k (v) = l∈Tk (x/m,y) aml v α(l) . We still have f m,k ∞ ≤ 1. Indeed, f m,k is in its turn a Fourier coefficient of f m : 1 f m (v) = (2π)



f m (veiθ )e−ikθ dθ.

T

Before embarking in the estimates, it is convenient to set Am,k =



|aml |,

Am =

l∈Tk (x/m,y)





Am,k =

|aml |.

(6.6.15)

l∈T (x/m,y)

k≥1

Now, the Wiener norm of P clearly verifies 

P W =

Am ≤ (x, y) sup

Am

(6.6.16)

m∈S(x,y)

m∈S(x,y)

and for m ∈ S(x, y) we proceed to the majorization of Am = this sum in two parts:

 k≥1

Am,k by splitting

(a) k ≤ 2R, where the parameter R is adjusted in order that the function k → (log √ x)/k + k log y =: h(k) attains its minimum at R, namely R = log x/ log y. Then, we have by Hölder, and using Theorem 6.6.6 (observe that 2k/(k + 1) = pk , with the notations of that theorem) as well as Lemma 6.6.8 and the fact that f m,k ∞ ≤ 1 (and increasing c if necessary):  Am,k ≤



l∈Tk (x/m,y)

|aml |

2k k+1

 k+1 2k

|Tk (x, y)|

√ k 1 k−1 ≤ C k x y − 2 x − 2k [ecy log2 x ] 2k

k−1 2k

6.6 A Sharp Sidon Constant

191

' ' & & √ √ 1 1 ≤ C 2R xecy log2 x exp − h(k) ≤ C 2R xecy log2 x exp − h(R) . 2 2 By summation, we therefore get T1 (m) :=



Am,k  RC

2R

√

xe

cy log2 x

k≤2R

& ' 1 exp − h(R) . 2

(b) k > 2R. Here, we use Cauchy-Schwarz, Theorem 6.2.5 and Lemma 6.6.8 to get T2 (m) :=





Am,k =

|aml | ≤



l∈∪k>2R Tk (x/m,y)

k>2R

|an |2

 21

1

N2R (x, y) 2

n≤x

1 2

1 2

1

= P 2 N2R (x, y) ≤ P ∞ N2R (x, y) = N2R (x, y) 2 √  x y −Recy log2 (x) . Now, we adjust the parameter y as big as possible to still have the factor ecy log2 negligible, namely we take √ log x , y= log2 (x)

y log2 (x) =



√

R = ( 2 + o(1))

log x,

x

* log x . log2 x

(6.6.17)

We then have 

 1 R log y = √ + o(1) λ(x) = (b0 + o(1))λ(x). 2 1 1 log x 1 h(R) = (R log y + ) = (2R log y) = R log y. 2 2 R 2 This gives us Am = T1 (m) + T2 (m) 

√

x exp [−(b0 + o(1))λ(x)] .

√ We obtain supm∈S(x,y) Am  x exp[−(b0 + o(1))λ(x)]. Now, observe that if α1 n = p1 · · · psαs ∈ S(x, y), we have 2α1 +···+αs ≤ n ≤ x. Since s ≤ y, this gives    log x s  exp(c log x). ψ(x, y) ≤ 1 + log 2 Finally, using (6.6.16), we obtain P W ≤ ψ(x, y) sup

Am 

√

x exp [−(b0 + o(1))λ(x)] ,

m∈S(x,y)

ending the proof of Theorem 6.6.7.



192

6 Hardy Spaces of Dirichlet Series

6.6.4 A Refined Inclusion As an application of the sharp upper bound obtained for the Sidon constant S( N ), here is a refinement of the inclusion H∞ ⊂ H2 , given by the following inequality [3], and which was improved in [39] concerning the optimal value of c: Theorem 6.6.9 (Balasubramanian–Calado–Queffélec) There exists a numerical  −s ∈ H∞ , we have constant C such that, if f (s) = ∞ n=1 an n ∞  |an | √ ≤ C f ∞ . n n=1

More precisely, if λ(n) = any 0 ≤ c < b0 :



(6.6.18)

log n log2 n, and if b0 is as in (6.6.12), we have for

∞  |an | cλ(n) ≤ C f ∞ . √ e n n=3

(6.6.19)

Proof Fix d, d  such that c < d < d  < b0 . We denote by a a numerical  constant −s which can vary and by μn the quantity μn = ecλ(n) . Let now f (s) = ∞ n=1 an n  Nk+1 ∞ k −s ∈ H , set Nk = 2 and f k (s) = n=1 an n . Then, we have, using (6.6.12) and then (6.2.1) and Nk+1 = 2Nk : 

|an | ≤

Nk ≤n 0.  −s is a fixed element of H∞ with 10. Throughout this exercise, f (s) = ∞ n=1 an n f ∞ = 1. (i) Let A be the matrix (on the Hilbert space C2 ):   a0 A= ba Show the equivalence A ≤ 1 ⇐⇒ |a|2 + |b| ≤ 1. (ii) Fix an integer q ≥ 2. Let M = 1, q −s  be the subspace of H2 generated by 1 and q −s , and T : M → M defined by T (h) = P( f h) where P : H2 → M is the orthogonal projection. Show that T ≤ 1 and that the matrix of T on the orthonormal basis (1, q −s ) is   a1 0 A= aq a1 Deduce that

|aq | ≤ 1 − |a1 |2 .

(iii) Show the following improvement in one of Bohr’s inequalities:  p

|a p | ≤ 1 − |a1 |2 .

References

195

References 1. H. Hedenmalm, P. Lindqvist, K. Seip, A Hilbert space of Dirichlet series and a system of dilated functions in L 2 (0, 1). Duke Math. J. 86, 1–36 (1997) 2. F. Bayart, Hardy spaces of Dirichlet series and their composition operators. Monat. Math. 136, 203–226 (2002) 3. R. Balasubramanian, B. Calado, H. Queffélec, The Bohr inequality for ordinary Dirichlet series. Studia. Math. 175, 285–304 (2006) 4. E. Saksman, Oslo special year Operator related function theory and Time-Frequency analysis (2012–2013), Private Communications 5. H. Bohr, Über die gleichmässige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913) 6. E. Landau, Über die gleichmässige Konvergenz Dirichletscher Reihen. Math. Zeit. 11, 317–318 (1921) 7. E. Saksman, K. Seip, Integral means and boundary limits of Dirichlet series. Bull. Lond. Math. Soc. 41(3), 411–422 (2009) 8. F. Bayart, S. Konyagin, H. Queffélec, Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series. Real. Anal. Exchange, 29(2), 557–586 (2003–2004) 9. J. Garnett, Bounded Analytic Functions, Revised First edn. (Springer, 2007) 10. B. Berndtsson, A. Chang, K.C. Lin, Interpolating sequences in the polydisk. Trans. Amer. Math. Soc. 302(1), 161–169 (1987) 11. H. Hedenmalm, E. Saksman, Carleson’s convergence theorem for Dirichlet series. Pacific. J. Math. 208, 85–109 (2003) 12. S. Konyagin, H. Queffélec, The translation 21 in the theory of Dirichlet series. Real Anal. Exchange (27), 155-175 (2001–2002) 13. E. Berkson, M. Paluszynski, G. Weiss, Transference Couples and Their Applications to Convolution Operators and to Maximal Operators, Lecture Notes in Pure and Applied Mathematics, vol. 175, pp. 69–84 (1996) 14. J.F. Olsen, K. Seip, Local interpolation in Hilbert spaces of Dirichlet series. Proc. Amer. Math. Soc. 136, 203–212 (2008) 15. J. Gordon, H. Hedenmalm, The composition operators on the space of Dirichlet series with square-summable coefficients. Michigan Math. J. 46, 313–329 (1999) 16. C. Finet, H. Queffélec, A. Volberg, Compactness of composition operators on a Hilbert space of Dirichlet series. J. Funct. Anal. 211, 271–287 (2004) 17. C. Finet, H. Queffélec, Numerical range of composition operators on a Hilbert space of Dirichlet series. Linear Algebra Appl. 377, 1–10 (2004) 18. F. Bayart, C. Finet, D. Li, H. Queffélec, Composition operators on the Wiener-Dirichlet algebra. J. Oper. Theory 60(1), 45–70 (2008) 19. H. Queffélec, K. Seip, Approximation numbers of composition operators on the H 2 space of Dirichlet series. Journ. Funct. Anal. 268, 1612–1645 (2015) 20. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres (Institut Elie Cartan, 1995) 21. B. Calado, Inégalité de Bohr pour les séries entières et les séries de Dirichlet et factorisation par convolution des fonctions continues périodiques. Thèse Université de Paris-Sud Orsay 2778 (2006) 22. J. Mac-Carthy, Hilbert spaces of Dirichlet series and their multipliers. Trans. Amer. Math. Soc. 356(3), 881–893 (2004) 23. A. Aleman, J.F. Olsen, E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series. Int. Math. Res. Not. IMRN 16, 4368–4378 (2016) 24. H. Helson, Hankel forms and random variables. Studia Math. 176(1), 85–92 (2006) 25. D. Vukotic, The isoperimetric inequality and a theorem of Hardy and Littlewood. Amer. Math. Monthly 110, 532–536 (2003) 26. F. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Analysis 37, 218–234 (1980)

196

6 Hardy Spaces of Dirichlet Series

27. A. Bonami, Etude des coefficients de Fourier des fonctions de L P (G). Ann. Inst. Fourier. Grenoble 20, 335–402 (1970) 28. H. Helson, Hankel forms. Studia Math. 198, 79–83 (2010) 29. O. Brevig, K.M. Perfekt, K. Seip, A.G. Siskakis, D. Vukotic, The multiplicative Hilbert matrix. Adv. Math. 302, 410–432 (2016) 30. P. Halmos, A Hilbert space problem book, 2nd edn. (Springer, 1982) 31. J. Ortega-Cerdà, K. Seip, A lower bound in Nehari’s theorem on the polydisc. J. Anal. Math. 118(1), 339–342 (2012) 32. S. Ferguson, M. Lacey, A characterization of products BMO by commutators. Act. Math. 189, 143–160 (2002) 33. M. Lacey, E. Terwilleger, Hankel operators in several complex variables and product BMO. Houston J. Math. 35, 159–183 (2009) 34. A.J. Harper, Moments of random multiplicative functions, I: low moments, better than squareroot cancellation, and critical multiplicative chaos. Forum Math. Pi 8, e1, 95pp (2020) 35. M. Weber, Private communication, Lille (2016) 36. A. Bondarenko, K. Seip, Helson’s problem for sums of a random multiplicative function. Mathematika 62(1), 101–110 (2016) 37. A.J. Harper, A. Nikeghbali, M. Radziwill, A note on Helson’s Conjecture on Moments of Random Multiplicative Functions. Analytic Number Theory (Springer, Cham, 2015), pp. 145– 169 38. B. Maurizi, H. Queffélec, Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 676–692 (2010) 39. A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaies, K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann Math. 174(1), 485–497 (2011) 40. R.C. Blei, Fractional cartesian products of sets. Ann. Inst. Fourier. Grenoble 29(2), 79–105 (1979) 41. K. Seip, Blei’s Inequality, Private Communication, Oslo special year operator related function theory and time-frequency analysis (2012–2013) 42. R. de la Bretèche, Sur l’ordre de grandeur des polynômes de Dirichlet. Acta Arithm. 134(2), 141–148 (2008) 43. M. Balazard, Remarques sur un théorème de G. Halàsz et A. Sàrközy. Soc. Math. France 117, 389–413 ( 1989) 44. M. Hüttner, On linear independence measures of some abelian integrals. Kyushu J. Math. 57(1), 129–157 (2003)

Chapter 7

Voronin-Type Theorems

7.1 Introduction 7.1.1 A Reminder About Zeta and L-Functions In this introductory section, we begin by fixing some notations, recalling some basic facts on Dirichlet characters [1, Chap. 5] and presenting the main results to be discussed. The techniques (Hilbertian spaces of analytic functions, ergodic theorems) are a good illustration of the material introduced in the previous chapters. P = { pk }k≥1 , p1 = 2, p2 = 3, . . . will denote the set of primes. If q is a positive integer, and if (Z/qZ)∗ is the multiplicative group of invertible integers mod q, a Dirichlet character mod q is a character χ (in the sense of abelian groups, as in Chap. 1) from (Z/qZ)∗ to T. The function χ is then extended to a completely multiplicative function χ : N∗ → C by the following formula: if n = mq + r is the Euclidean division of n by q, with 0 ≤ r < q, one sets χ(n) = χ(r ) if (r, q) = 1; χ(n) = 0 if (r, q) > 1. This function χ is of period q. The principal character χ0 mod q is defined by χ0 (n) = 1 if (n, q) = 1; χ0 (n) = 0 if (n, q) > 1. Although we will not really use it, let us now define the important notion of primitivity for a character χ mod q. If d is a positive integer, and j : (Z/dqZ)∗ → (Z/qZ)∗ the natural map, then χ1 = χ ◦ j is a character mod Q = dq, which is said to be induced by the character χ mod q, where q|Q. Reversing things, we will say that a character χ mod q is primitive if it is induced by no character χ∗ mod q ∗ , where q ∗ is a proper divisor of q. Any character χ is induced by a unique primitive character χ∗ mod q ∗ , where q ∗ is a divisor of q. And q ∗ is called the conductor of χ. © Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_7

197

198

7 Voronin-Type Theorems

If χ is a Dirichlet character mod q ≥ 1, the associated L-function is [1, p. 31] L χ (s) = L(s, χ) =

∞ 



χ(n)n −s =

(1 − χ( p) p −s )−1 , Re s > 1.

p

n=1

If q ∗ is the conductor of a character χ mod q, the Euler products associated with χ and χ∗ are connected by L(s, χ) = L(s, χ∗ )



[1 − χ ( p) p −s ].

(7.1.1)

p|q

The case q = 1 and χ(n) = 1 is that of the Riemann zeta function: ζ(s) =

∞ 

n −s =



(1 − p −s )−1 , Re s > 1.

p

n=1

If = χ0 , the Dirichlet series of L(s, χ) has a convergence abscissa σc = 0, since χ q n=1 χ(n) = 0. If χ = χ0 , we have σc = 1 and L(s, χ0 ) = ζ(s)



(1 − p −s ).

(7.1.2)

p|q

All those functions have a meromorphic extension to the complex plane, with at most a simple pole at s = 1 [1, pp. 69–71] when χ is principal. Let us set τ (χ) =

q 

m

χ(m)e2iπ q .

m=1

This is a Gaussian sum, whose modulus verifies |τ (χ)| = We introduce the ξ-function defined by ξ(s) =

√ q for χ = χ0 . [1, p. 66].

s 1 s(s − 1)π −s/2 ( )ζ(s). 2 2

We also set α = 0 if χ(−1) = 1, α = 1 if χ(−1) = −1 and then we define the twisted ξ-function   s+α − 21 (s+α) L(s, χ). ξ(s, χ) = (π/q)  2 With those notations, the functional equations respectively verified by ζ and the L-functions, in the case of a primitive character, are

7.1 Introduction

199

ξ(1 − s) = ξ(s), √ iα q ξ(s, χ), χ = χ0 . ξ(1 − s, χ) = τ (χ)

(7.1.3) (7.1.4)

As a consequence of those functional equations, of the Stirling formula for , and of the relation (7.1.1), denoting by f either the zeta function or an arbitrary Dirichlet L-function, we have σ ≤ −δ < 0 and |t| > 1 =⇒ f (σ + it) = O(|t|1/2−σ )

(7.1.5)

where the O is uniform for fixed δ. Those consequences are derived in detail in [1], Chaps. 8 and 9, or in [2, pp. 146–147], which uses the form  ζ(s) = ρ(s)ζ(1 − s) with ρ(s) = 2(2π)s−1 (1 − s) sin

 1 πs . 2

A more elementary version [1, Chap. 12] is as follows. Theorem 7.1.1 For s = σ + it with σ > 1/2, |s − 1| ≥ δ > 0, and χ non-principal mod q, we have |ζ(s)| ≤ C|s| and |L(s, χ)| ≤ 2q|s|. This is indeed a simple consequence of the formulas  ∞ s −s ζ(s) = (x − [x])x −s−1 d x, σ > 0 s−1 1  ∞  L(s, χ) = s S(x)x −s−1 d x, S(x) = χ(n), |S(x)| ≤ q. 1

n≤x

We will also need quadratic mean estimates, which are more involved as follows. Theorem 7.1.2 For s = σ + it with 1/2 < α ≤ σ ≤ β < 1, and χ a Dirichlet character mod q, we have for some C = C(α, β, q) : 1 T



T

1 |ζ(σ + it)| dt ≤ C and T



T

2

0

|L(σ + it, χ)|2 dt ≤ C.

0

The case of the zeta function is treated in detail in [3, pp. 77 and 140]. A very complete and clear proof can also be found in [4, pp. 269–271 and 277], or in [5, p. 21]. All proofs are on the following pattern. First, one establishes an “approximate formula”. Then, one derives more or less easily from that formula an L 2 -estimate. We indicate two possible approaches:

200

7 Voronin-Type Theorems

1. The proof of the approximate formula mentioned here is based on a Kuzmin– Landau-type theorem for oscillatory integrals or sums [3, p. 77]. Let us quote without proof that formula for ζ. Theorem 7.1.3 For x ≥ 1 and s = σ + it, we have ζ(s) =



n −s −

n≤x

x 1−s + O(x −σ ) 1−s

uniformly for σ ≥ σ0 > 0 and 0 < t ≤ 2πx/C, where C is a given constant > 1. In particular (on taking x = t), we have, uniformly for 0 < σ0 ≤ σ ≤ σ1 < 1 and for t →∞:  n −σ−it + O(t −σ ). (7.1.6) ζ(σ + it) = n≤t

The relation (7.1.6) applies as well to the Hurwitz zeta function ζ(s, θ) =

∞ 

(n + θ)−s , 0 < θ ≤ 1.

n=0

(Note that ζ(s) = ζ(s, 1)). One just has to replace, in the oscillatory exponential sum N 

n −it =

n=1

N 

e2iπ f (n) with f (x) = −(2π)−1 t log x

n=1

implied in the proof, the function f by f (x) = −(2π)−1 t log(x + θ). Now, whether χ = χ0 or not, one has by Euclidean division, due to the periodicity of χ: L(s, χ) =

 m≥0, 1≤r ≤q

χ(r )(mq + r )−s = q −s



χ(r )ζ(s, r/q),

1≤r ≤q

so that (7.1.6) applies to L-functions as well, and that, more precisely and for fixed q, L 2 -estimates for ζ(s, θ) will imply L 2 -estimates for L(s, χ). The derivation of those L 2 -estimates will be detailed in the second approach (recall in passing that L(s, χ) is an entire function for χ = χ0 , as was proved in Chap. 3, after Theorem 2.5). 2. An alternate proof of Theorem 7.1.3, in the spirit of Ramachandra’s work, was kindly indicated to me by Ramaré [6]. It avoids the appeal to oscillatory exponential  −s a denote either ζ or a Dirichlet sums and works as follows: let f (s) = ∞ n=1 n n L-function. We will set c f = 1 if f = ζ or L(s, χ0 ) and c f = 0 if f = L(s, χ) with χ non-principal. Then, we have the following. Proposition 7.1.4 Let 1/2 < α − δ < α ≤ σ0 ≤ β < 1, x ≥ 1 and s0 = σ0 + it, with t ≥ 0. Then, with uniform estimates for fixed α, β, δ :

7.1 Introduction

201

f (s0 ) =

  c f x 1−s0 n an n −s0 − 1− x (1 − s0 )(2 − s0 ) 1≤n≤x

 + O (1 + t)1/2+δ x −σ0 −δ .

(7.1.7)

In particular ,  1−σ0    n x −σ0 −it f (σ0 + it) = an n 1− +O x 1 + t2 1≤n≤x

 + O ((1 + t)/x)1/2+δ .

(7.1.8)

Proof We first note that, as a consequence of the Perron–Landau formulas of Chap. 4, one has  2+i∞ ys 1 ds = 0 (7.1.9) 0 < y < 1 =⇒ 2iπ 2−i∞ s(s + 1)  2+i∞ 1 ys ds = 1 − 1/y. (7.1.10) y > 1 =⇒ 2iπ 2−i∞ s(s + 1) Indeed, using 1/s(s + 1) = 1/s − 1/(s + 1) and shifting, one sees that (the symbol c+i∞ c+i T c−i∞ meaning as usual lim T →+∞ c−i T ) 1 2iπ



2+i∞ 2−i∞

1 ys ds = s(s + 1) 2iπ



2+i∞

1 ys ds − s 2iπ y

2−i∞



3+i∞ 3−i∞

ys ds s

and the Perron–Landau formulas [7, p. 342] or [2, p. 135] give the result. Secondly, using the estimate (7.1.5) and the Cauchy residue formula, we have, with now absolutely convergent integrals due to the presence of the weight 1/s(s + 1): 1 2iπ



2+i∞

f (s + s0 )

2−i∞

1 xs ds = s(s + 1) 2iπ



−σ0 −δ+i∞

−σ0 −δ−i∞

f (s + s0 )

xs ds s(s + 1)

c f x 1−s0 + f (s0 ) + . (1 − s0 )(2 − s0 ) Now, using (7.1.10) and (7.1.9) with y = x/n, the integral in the LHS equals ∞  n=1

an n

−s0

1 2iπ



2+i∞

2−i∞

 (x/n)s ds = (1 − n/x)an n −s0 . s(s + 1) 1≤n≤x

And, using (7.1.5) with s = −σ0 − δ + iτ , s0 = σ0 + it and Re(s + s0 ) = −δ, the integral I in the RHS is evaluated as follows, since 1/2 + δ < 1 and |t + τ |1/2+δ ≤ |t|1/2+δ + |τ |1/2+δ :

202

7 Voronin-Type Theorems

I x

−σ0 −δ

 R

|t + τ |1/2+δ dτ  x −σ0 −δ (1 + |t|)1/2+δ . τ2 + 1

To finish, we just have to note that |(1 − s0 )(2 − s0 )| 1 + t 2 and σ0 + δ ≥ 1/2 + δ. 

This ends the proof.

As a consequence (this would apply to (7.1.6) as well), we shall prove the following. Proposition 7.1.5. If f = ζ or f = L, we have, in a uniform manner for 1/2 < α ≤ σ0 ≤ β < 1 :  T | f (σ0 + it)|2 dt = O(T ). (7.1.11) 0

Proof. In [3, p. 140], this is derived from (7.1.6) in a rather technical way, in which the size of the sum depends on the variable t. Starting from (7.1.8), we shall give here a simpler and less technical proof, in which the size of the sum no longer depends on the variable. This proof is based on the generalized Hilbert inequality of Chap. 1 which we recall as  T 

2 

bn n −it dt  |bn |2 (T + n). (7.1.12)

0

n

n

We take bn = an n −σ0 (1 − n/T )+ and we use (7.1.8) when x is equal to T , getting for 0 ≤ t ≤ T , due to the inequality (1 + t)/T ≤ 2 for T ≥ 1:



2 T 2(1−σ0 )

| f (σ0 + it)2  bn n −it + + O(1) (1 + t 2 )2 n≥1 =: Z 1 (t) + Z 2 (t) + O(1). We now use (7.1.12) to obtain, since |bn | ≤ |an |n −σ0 ≤ n −σ0 and σ0 ≥ α > 1/2: 

T

Z 1 (t)dt  T

0

∞ 



n −2σ0 +

n=1

n 1−2σ0  T + T 2−2σ0  T

1≤n≤T

and similarly 

T

Z 2 (t)dt  T 2(1−σ0 )

0

which ends the proof.



∞

(1 + t 2 )−2 dt  T 2(1−σ0 )  T,

0



7.1 Introduction

203

7.1.2 Universality Let us first coin some additional notations. H (U ) will be the space of holomorphic functions on an open set U of the complex plane. The critical strip , the set H ∗ () and the incomplete functions ζ N and L N are defined as 1 < Re s < 1} 2 H ∗ () = { f ∈ H (); f has no zeros in } N ζ N (s) = (1 − pk−s )−1 k=1 N L N (s, χ) = (1 − χ( pk ) pk−s )−1 .

 = {s ∈ C;

k=1

The Riemann hypothesis claims that ζ ∈ H ∗ (), as well as L(s, χ). There is a natural semi-group acting on the set (H ())r of all r -tuples f = ( f 1 , . . . , fr ) of H (), namely the semi-group of vertical translations: Tt f (s) = f (s + it), s ∈ , t ≥ 0. Recall (see Chap. 2) that the lower density of a (measurable) set D ⊂ R+ is defined by  1 T dens(D) = lim inf 1 D (t)dt. (7.1.13) T →∞ T 0 The upper density dens(D) is defined similarly. If the limit exists, we write dens(D). A family F of functions of H () will be called universal if the following holds: for any f ∈ H ∗ (), any compact subset K of  and any ε > 0, there is a vertical translate g(s + it) of some function g ∈ F such that sup | f (s) − g(s + it)| ≤ ε.

(7.1.14)

s∈K

Similarly, a family F of r -tuples of H () will be called jointly universal if, for any r -tuple ( f 1 , . . . , fr ) of functions in H ∗ (), any compact subset K of  and any ε > 0, there is a vertical translate of an r -tuple g = (g1 , . . . , gr ) ∈ F such that sup sup | f j (s) − g j (s + it)| ≤ ε.

(7.1.15)

1≤ j≤r s∈K

Often, the family F of functions or of r -tuples will be a singleton {g}. To see how little room we have with vertical translates of the single function g, observe that, if this function is bounded in , say by C, there is no way of realizing (7.1.14). Simply take f (s) = C + 1. For example, the Dirichlet series

204

7 Voronin-Type Theorems

g(s) =

∞ 

2−ks = (1 − 2−s )−1 ,

k=0

with |g (s) | ≤ g(1/2) = 2 +

√ 2 throughout 

is not universal. The next two key results of this chapter are all the more striking. Theorem 7.1.6. (Voronin) Let f ∈ H ∗ (), ε > 0, K ⊂  be a compact set. Then, there exists a positive real number t such that sup | f (s) − ζ(s + it)| ≤ ε.

(7.1.16)

s∈K

Moreover, the set of all such t’s has positive lower density. This theorem admits a stronger version, with joint universality. Theorem 7.1.7. (Bagchi) Let q ≥ 1 be an integer and χ1 , . . . , χr be distinct characters mod q. Let f 1 , . . . , fr ∈ H ∗ (), ε > 0, K ⊂  be a compact set. Then, there exists a positive real number t such that

sup sup f j (s) − L(s + it, χ j ) ≤ ε.

(7.1.17)

1≤ j≤r s∈K

Moreover, the set of all such t’s has positive lower density. Remark. The first striking result along these lines was obtained by Bohr in 1915 [8]: for fixed σ ∈ (1/2, 1), the curve t → ζ(σ + it), where t ∈ R, has dense range in the complex plane. Now, those universality theorems, which considerably reinforce Bohr’s initial statement, say that the vertical translates of ζ or of r -tuples of Lfunctions approximate anything that they can reasonably approximate if perchance Riemann’s hypothesis holds true (recall that, according to the Hurwitz theorem, a locally uniform limit of analytic functions without zeros is either analytic without zeros or identically zero). The zeta function or L-functions thus appear as explicit universal objects (the fact that they admit Euler products playing probably a key role, although this is not absolutely needed; see exercise 5). A remarkable fact is that the existence of such universal objects (like hypercyclic vectors; see, e.g.. chapter one of [4]) is generally proved by the use of the Baire category theorem, so that the implied objects are by no means explicit. The only other example the authors are aware of is that of the Riemann (again him!) function R(x) =

∞  sin n 2 πx n=1

n2

which verifies the “multifractal formalism” of Frisch and Parisi ([9]).

7.1 Introduction

205

Coming back to Theorem 7.1.6 or 7.1.7, of course, the second one contains the first. But already in the case of the Riemann function, there are formidable difficulties! And the reader can at first reading consider that r = 1 and χ j ( p) ≡ 1. The strategy of the proof, summarized for the ζ function, is now well-established [10], and consists of three steps, for a detailed exposition of which we have to acknowledge that we owe very much to the beautiful presentation given in [4] by Bayart and Matheron. (1) One first proves a Hilbertian approximation result for an arbitrary function f ∈ H () by unimodular combinations of the “monomial” functions f p (s) = p −s . (2) Moreover, if f ∈ H ∗ (), we can exponentiate the first approximation, and obtain a locally uniform approximation of f by finite Euler products with unimodular coefficients. Using the Kronecker theorem, one even obtains locally uniform approximation of f by many translates of some incomplete zeta function ζ N . This is an interesting result in itself, since it shows (in a strong sense) the universality of the sequence of functions (ζ N ). (3) We are more ambitious and want to replace the sequence (ζ N ) by the single function ζ. Here, still more formidable technical difficulties appear, all the more as we are in the critical strip, where in principle no reasonable approximation of ζ by the ζ N ’s is to be expected. We shall circumvent this difficulty by the use of three ingredients: a general Carlson identity, difficult to prove, but interesting in itself; the Birkhoff–Khinchin theorem seen in Chap. 2; and finally the appeal to some Hilbert space (a Bergman space) of analytic functions.

7.2 Hilbertian Results 7.2.1 A Hilbertian Density Criterion The main theorem of this subsection is the following. Theorem 7.2.1. Let (xn )n≥1 be a sequence of vectors of a Hilbert space H such that : ∞ ∗ ∗ ∗ (1) ∀x ∞∈ H, x2 = 0, we have n=1 |x (xn )| = ∞ (2) n=1 xn  < ∞.  Then, for any positive integer N0 , the set K = { rn=N0 an xn ; |an | = 1} of all remote, finite, unimodular, combinations of the vectors xn is dense in H . Proof. First, a word on the assumptions. The first one is a condition of big size on the xn , the second one a condition of small size. These two seemingly antinomic conditions will ensure that the unimodular combinations, not all linear are normones, ∞ x dense in H . Besides, the first condition obviously implies that n  = ∞. n=1  Then, still assuming n xn 2 < ∞, a theorem of Drobot [11] claims the following: assume that the subspace

206

7 Voronin-Type Theorems

X := {x ∈ H ;

∞ 

| x, xn  | < ∞}

n=1

is closed in H and let Y = X ⊥ . Then, there exists s0 ∈ H such that the set of convergent rearrangements of the series ∞ n=1 x n is exactly s0 + Y . If dim H < ∞, the assumption is automatically fulfilled, and the theorem was proved by Steinitz [12]. It is interesting to compare those results with the present one, which assumes X = {0}. Now to the proof, for which we may assume that N0 = 1 since the sequence (xn )n≥N0 verifies the same assumptions as the whole sequence. This proof will consist of two steps: Step 1. Fix an integer N and denote by K N the following set: KN =



 an xn ; |an | ≤ 1 , where the sums are finite.

n>N

Then, K N is dense in H . Indeed, K N is convex and balanced in H (this is why we need to allow sub-unimodular coefficients). Therefore, by the Hahn–Banach theorem (here actually the projection theorem), it is enough to show that, for any x ∈ H , we have



(7.2.1) ∀x ∗ ∈ H \{0}, x ∗ (x) ≤ sup x ∗ (y) . y∈K N

 Let M > N be such that N 0 can be written as (using the Prime Number theorem with remainder for arithmetic progressions; see [21, p. 381] n j = πq,b (ew j +α j ) − πq,b (ew j )  ew j +α j √ 1 dt = + O(ew j +α j e−c w j +α j ) w ϕ(q) e j log t √ ew j (eα j − 1) ≥ + O(ew j +α j e−c w j +α j ), ϕ(q)(w j + α j ) where ϕ is the Euler totient function. But we know that w j = x j + O(1) and that α j ≈ 1/x 4j , so that for some constant ρ > 0: n j ≥ ρe x j /x 5j ,

j ≥ j0 .

The relation (7.2.14) now implies, for j ≥ j0 :  log p∈I j p∈Pq,b

1 eδx j | f (log p)| ≥ n j e−(1−δ)x j ≥ ρ 5 → ∞, 4 4x j ,

and that ends the proof of Proposition 7.2.6.



7.2 Hilbertian Results

213

We will next check that (7.2.9) holds. Let  be a non-zero element of H = (B 2 (U ))r , given by r -tuple (1 , . . . , r ) of functions in B 2 (U ) acting on g = (g1 , . . . , gr ) ∈ H by the formula , g =

r   j=1

In particular, 



, f p =

r 

 j (w)g j (w)d A(w).

U

 χ j ( p)

 j (w) p −w d A(w).

U

j=1

For an integer b coprime with q, it will be convenient to set Fb (z) =

r 

 χ j (b)

j=1

Fb (z) =

 j (w)e−wz d A(w), or

U

  ∞ r  (−1)k z k k=0

k!

χ j (b) j (w)w k d A(w).

(7.2.15)

U j=1

The function Fb is clearly entire with exponential type. Moreover, using the periodicity χ j ( p) = χ j (b) for p ∈ Pq,b , we can write     | , f p | =



|Fb (log p)|.

(7.2.16)

(b,q)=1 p∈Pq,b

p

We shall first prove the following lemma. Lemma 7.2.9. Fix b coprime with q, and set β = supw∈U Re w < 1. Then, if r  (w)χ j (b) = 0 for at least one w, we have as follows : j j=1 lim sup x→+∞

log |Fb (x)| ≥ −β. x

(7.2.17)

Indeed, if the conclusion fails, we can find C, δ > 0 such that x ≥ 0 =⇒ |Fb (x)| ≤ Ce−(β+2δ)x .

(7.2.18)

Now, we see that (increasing C if necessary) x < 0 =⇒ |Fb (x)| ≤

r   j=1

U

e−xRe w | j (w)|d A(w) ≤ Ceβ|x| .

(7.2.19)

214

7 Voronin-Type Theorems

Setting G(z) = e(β+δ)z Fb (z), both estimates imply |G (x)| ≤ Ce−δ|x| for all x ∈ R. Therefore, G is an entire function of exponential type, whose restriction to the real line is L 2 . By the Paley–Wiener theorem [16, p. 375], its Fourier transform  = G(ξ)



G(x)e−iξx d x R

is supported by (−A, A) for some real number A > 0. But |G(x)e−i xξ | ≤ Ce−(δ−|Im ξ|)|x| ,  is analytic in the strip {|Imξ| < δ}, and vanishes on [A, ∞[. This implies therefore G  = 0. By the unicity theorem for the Fourier transform, we get as well G = 0 and G  Fb = 0, i.e. U rj=1 χ j (b) j (w)w k d A(w) = 0 for k = 0, 1, . . . by (7.2.15). Now, the polynomials are dense in B 2 (U ) by Theorem 7.2.3. We thus get   r

χ j (b) j (w)h(w)d A(w) = 0 for all h ∈ B 2 (U ).

U j=1

 This implies rj=1 χ j (b) j (w) = 0 for all w ∈ U , a contradiction.    Finally, we  verify (7.2.9). Suppose that p | , f p | < ∞. Then, using (7.2.16), we see that p∈Pq,b |Fb (log p)| < ∞ for each b coprime with q. By Lemmas 7.2.6  and 7.2.9, we get rj=1 χ j (b) j (w) = 0 for all b, w. Now the characters χ1 , . . . , χr , considered as morphisms of the group G = (Z\qZ)∗ to C∗ are linearly independent, by orthogonality or by Dedekind’s lemma [22, p. 47]. Therefore,  j (w) = 0 for all j, w, implying  = 0, and finishing the proof of (7.2.9) by contradiction. 

7.3 Joint Universality of the Sequence (λ N ) So far, even though primes and the critical strip were mentioned, the zeta function or a L-function did not explicitly appear. This will be the case in this section. For χ1 , . . . , χr distinct characters mod q (with r ≤ ϕ(q)), we set j

L N (s) = L N (χ j , s) =

N  k=1

(1 − χ j ( pk ) pk−s )−1 and λ N = (L 1N , . . . , L rN ).

7.3 Joint Universality of the Sequence (λ N )

215

More generally, for z = (z k ) ∈ T∞ , we will set j L N (s, z)

=

N 

(1 − z k χ j ( pk ) pk−s )−1

(7.3.1)

k=1

and define the r -tuple (λzN ) by λzN = (L 1N (., z), . . . , L rN (., z)). Recall that f p = ( f p1 , . . . , f pr ) with

f pj (s) = χ j ( p) p −s .

In this second step, we prove the joint universality of the sequence (λ N ). First, with the notations of (7.2.7), we have the following form of the approximation result. Proposition 7.3.1. Let f 1 , . . . , fr ∈ H ∗ (), ε > 0, K ⊂  a compact set and N0 ∈ N. Then, there exists an integer N ≥ N0 and unimodular complex numbers z 1 , . . . , z N such that N 



 f − λzN Y = sup sup f j (s) − 1≤ j≤r s∈K

k=1

1

−s ≤ ε. 1 − z k χ j ( pk ) pk

(7.3.2)

Proof. We set σ0 = inf s∈K Re s > 1/2. For s ∈  and z ∈ T, let h k, j (s, z) =

∞  n=1

χ j (npk )

pk−ns z n , with eh k, j (s,z) = (1 − zχ j ( pk ) pk−s )−1 . n

We observe that, for s ∈ K and z ∈ T: |h k, j (s, z) − χ j ( pk )zpk−s | ≤

∞ 

pk−nσ0 ≤ C pk−2σ0 .

(7.3.3)

n=2

Since  is simply connected, we can write f j = eg j with g j ∈ H (). Let L be the compact set D(0, gY ) where g = (g1 , . . . , gr ) ∈ Y := (C(K ))r . By uniform continuity, we can find δ = δ(ε) > 0 such that

if a ∈ L and b ∈ C, then |a − b| ≤ 2δ =⇒ ea − eb ≤ ε.

(7.3.4)

Let now U be a smooth Jordan domain (e.g. the inside of an ellipse) such that K ⊂U ⊂U ⊂

216

7 Voronin-Type Theorems

and N0 an integer to be adjusted later. Let also G j (s) = g j (s) −

N0 

h k, j (s, 1), 1 ≤ j ≤ r.

k=1

 Since, by Theorem 7.2.5, the unimodular sums k≥N0 z k f pk are dense in the space H = (B 2 (U ))r whose norm dominates that of (C(K ))r by (7.2.8), we can find N ≥ N0 and unimodular numbers z N0 +1 , . . . , z N such that N



χ j ( pk )z k pk−s ≤ δ. sup sup G j (s) −

1≤ j≤r s∈K

k=N0

Setting z 1 = · · · = z N0 = 1, this implies, for s ∈ K and thanks to (7.3.3):



N N  



h k, j (s, z k ) = G j (s) − h k, j (s, z k )

g j (s) − k=1 k=N0 +1

N N  

−s ≤ G j (s) − χ j ( pk )z k pk + |χ j ( pk )z k pk−s − h k, j (s, z j )| k=N0 +1 N0 +1−2σ0 pk . ≤δ+C k>N0

We can now adjust N0 so as to have N



sup sup g j (s) − h k, j (s, z k ) ≤ 2δ.

1≤ j≤r s∈K

Using (7.3.4) with a = g j (s) and b =

k=1

N k=1

h k, j (s, z k ), we now obtain

N 



sup sup f j (s) −

1

−s 1 − χ ( p )z p 1≤ j≤r s∈K j k k k k=1

N

g j (s)

h (s,z ) k, j k = sup sup e − e k=1

≤ ε. 1≤ j≤r s∈K

And this ends the proof of Proposition 7.3.1.



j

Now, with our previous notations χ j , L N , we have the following theorem, already very interesting in itself. Theorem 7.3.2. Let f 1 , . . . , fr ∈ H ∗ (), ε > 0, K ⊂  a compact set, N0 a positive integer. Then, there exists an integer N ≥ N0 , and a set D ⊂ R+ of positive density such that, for every t ∈ D, one has

7.3 Joint Universality of the Sequence (λ N )

217 j

 f − Tt λ N Y = sup sup | f j (s) − L N (s + it)| ≤ ε.

(7.3.5)

1≤ j≤r s∈K

 k Proof. Denote by log(1 − z) the series − ∞ k=1 z /k, |z| < 1. First, using Proposition 7.3.1, we approximate the f j ’s uniformly on K with an error ≤ ε/2 N  1 by a product k=1 1−χ ( p )z p−s , N ≥ N0 . For w ∈ T∞ , s ∈ K , 1 ≤ j ≤ r and j k k k some constants C N , C N depending only on N , we can write N N





−s −1 (1 − χ ( p )z p ) − (1 − χ j ( pk )wk pk−s )−1

j k k k k=1

k=1

N N

   

= exp − log(1 − χ j ( pk )z k pk−s ) − exp − log(1 − χ j ( pk )wk pk−s k=1

≤ CN

k=1 N 



log(1 − χ j ( pk )z k pk−s ) − log(1 − χ j ( pk )wk pk−s )

k=1

≤ C N

N 

|z k − wk |.

k=1

N In other terms, we have λzN − λwN Y ≤ C N k=1 |z k − wk |. Now, by Proposition 7.3.1, we can find N ≥ N0 and z ∈ T∞ such that λzN − f Y < ε/2. Let V ⊂ T∞ the open set defined by V = {w ∈ T∞ ; sup |wk − z k | < δ}.

(7.3.6)

1≤k≤N

By the above, if δ ≤ ε/(2N C N ), we have V ⊂ {w; λwN − f Y < ε}, the set V is open (only the first N coordinates are restricted) and has therefore positive Haar measure μ(V ) > 0 in T∞ . Let now   D = t ∈ R+ ; K t e ∈ V , where

(7.3.7)

e = (1, . . . , 1, . . .) and K t e = ( p1−it , . . . , pk−it , . . .).

The Kronecker theorem of Chap. 2 tells us that D has positive density equal to μ(V ). Since λ KN t e = Tt λ N , we have by definition t ∈ D =⇒  f − Tt λ N Y < ε which ends the proof.



218

7 Voronin-Type Theorems

7.4 A Generalized and Uniform Carlson Formula 7.4.1 Estimates on the Gamma Function 7.4.1.1

Vertical Estimates

Actually, the estimates on  were already used in the first section, with the functional equations and the approximate formulas... we now need them again, under a more specific form. Recall [23, Chap. 1] or [21, Appendix C, p. 520] that the gamma function  of Euler is meromorphic in the whole complex plane with simple poles at 0, −1, . . . , −k, . . ., and verifies (1) = 1 and the two functional equations (s + 1) = s(s) and (s)(1 − s) =

π . sin πs

As a consequence of the (iterated) first functional equation, one has Res (, −k) =

(−1)k , k = 0, 1, . . . k!

(7.4.1)

The basic fact on the gamma function is Stirling’s formula in the complex plane (see [21], Appendix C, p. 523), namely, (s) =

√ 2πs s−1/2 e−s (1 + O|s|−1 ))

(7.4.2)

which holds uniformly in every domain remote from the negative real axis: Dε = {s ∈ C; |s| > ε and |arg s| ≤ π − ε} ,

0 < ε < π.

As a first consequence of (7.4.2), we have the following (see, e.g. [23, p. 21]: Proposition 7.4.1. Let I ⊂ R be a segment which contains no non-positive integer. Then, we have σ ∈ I =⇒ |(σ + it)| ≤ K I e−ρ|t| for all t ∈ R

(7.4.3)

where K I < ∞ and ρ = π/4.

7.4.1.2

Horizontal Estimates

In complement to (7.4.3), we will also need the following. Proposition 7.4.2. Let T be a real number ≥ 2. Then, we have the upper bound (uniform for |σ| ≤ T )

7.4 A Generalized and Uniform Carlson Formula

219

   |(σ + i T )|  exp σ log σ 2 + T 2 − T arg(σ + i T ) ,

(7.4.4)

in which arg denotes the principal determination (−π < arg < π) of the argument. This proposition is a straightforward consequence of the Stirling formula (7.4.2) for . Just use the fact that, if s = σ + i T ,  |s s | = exp[Re(s log s)] = exp[σ log σ 2 + T 2 − T arg(σ + i T )], with 0 ≤ arg(σ + i T ) ≤ 3π/4 for |σ| ≤ T .

7.4.1.3

Mellin’s Inversion Formula for the Exponential

The proof of Carlson’s formula will start from the following identity [21, Appendix C, p. 525]. Proposition 7.4.3. Let a, b be positive real numbers. Then, we have the Mellin inversion formula  b+i∞ 1 −a e = (w)a −w dw. (7.4.5) 2iπ b−i∞ Proof. The idea is simple, but the details are fairly technical. Let T = N + 1/2 be a large half-integer and let C T the rectangle with vertices b ± i T and −T ± i T . The residue theorem gives (recall that, from (7.4.1), we have Res (, −k) = (−1)k /k!) (see Fig. 7.1 below) 2iπ

N  (−a)k

k!

k=0



=  +

b+i T

= 2iπ

N 

Res ((w)a −w , −k) =

k=0

(w)a −w dw+

b−i T −T −i T

−T +i T





(w)a −w dw

CT −T +i T

(w)a −w dw

b+i T

(w)a −w dw +



b−i T −T −i T

(w)a −w dw =:

4 

I j (T ).

j=1

b We have I2 (T ) = − −T (x + i T )a −x−i T d x, and the estimate (7.4.4) shows that (i) If 0 ≤ x ≤ b, we have |(x + i T )a −(x+i T ) |  exp[b log R − R arctan(R/b)]. (ii) If −T ≤ x ≤ 0, we have π/2 ≤ arg(x + i T ) ≤ 3π/4, so that, as soon as T ≥ a, we have  π  |(x + i T )a −(x+i T ) |  exp[x(log x 2 + T 2 − log a)] exp − T 2  π  exp − T . 2

220

7 Voronin-Type Theorems

Fig. 7.1 Backward shift

This shows that lim T →∞ I2 (T ) = 0. Since (x − i T ) = (x + i T ), we have lim T →∞ I4 (T ) = 0 as well. It remains to show that lim T →∞ I3 (T ) = 0, with  I3 (T ) = −i

T −T

(−T + it)a T −it dt.

Here, we can no longer use directly (7.4.4) since we are approaching the forbidden negative real axis. Yet, we shall see that (recall that T = N + 1/2 is a half-integer), for some constant δ > 0, we have |(−T + it)|  e−δT log T for |t| ≤ T.

(7.4.6)

Indeed, the complement formula for  gives



π

= π ≤ π.

|(−T + it)(1 + T − it)| = sin[π(−N − 1/2 + it)] ch πt Besides, |(1 + T − it)| = |(T − it)(T − it)| ≥ |(T − it)| and if s = T − it with |t| ≤ T , the Stirling formula (7.4.2) is available for (s), with

7.4 A Generalized and Uniform Carlson Formula

221

 t Re(s log s) = T log t 2 + T 2 − t arctan( ) T π ≥ T log T − T ≥ 2δT log T. 2 This gives |(s)| eδT log T , implying (7.4.6) and by the way, the fact that lim T →∞ I3 (T ) = 0. Passing to the limit in the residue formula as T → ∞, we thus get  b+i∞ 2iπe−a = lim I1 (T ) = (w)a −w dw, T →∞

b−i∞

which ends the proof of the Mellin inversion formula (7.4.5).



7.4.2 The Carlson Formula Recall that Cθ denotes the half-plane {s; Re s > θ}. Let f be analytic in Cα and σ > α. We will say that f is of finite order in Cσ if we have, for some positive constants A, B: | f (x + it)| ≤ A + |t| B throughout Cσ . With those notations, we have the following theorem.  −s with σc ( f ) < ∞, and let α, β ∈ Theorem 7.4.4. (Carlson) Let f (s) = ∞ n=1 an n R with α < β. Set α,β := {s; α < Re s < β}. Suppose that f has an analytic extension to Cα which verifies an L ∞ and an L 2 -estimate, namely, (a) For any σ > α, f is of finite order in Cσ . (b) For all compact sets I ⊂]α, β[, one has 1 T

sup sup

σ∈I T >0



T

| f (σ + it)|2 dt < ∞.

0

Then, we have a uniform Carlson identity : s ∈ α,β =⇒ lim

T →∞

1 T



T

| f (s + it)|2 dt =

0

∞ 

|an |2 n −2Re s ,

(7.4.7)

n=1

with uniform convergence on compact subsets of α,β . Proof. We follow [4, pp. 273–277], with some modifications. Let K ⊂ α,β be a compact set. We set σ0 = inf s∈K Re s > α and take b > 0 so large that b > σa ( f ) − σ0 . We introduce the mollified function gδ (s) =

∞  n=1

an e−nδ n −s , 0 < δ < 1.

(7.4.8)

222

7 Voronin-Type Theorems

Since σc ( f ) < ∞ implies that an is of polynomial growth, this is an entire function. We have an integral representation for that function at any point s = σ + it ∈ K : 1 gδ (s) = 2iπ



b+i∞

(w) f (s + w)δ −w dw.

(7.4.9)

b−i∞

Indeed, using (7.4.5), we can write gδ (s) =

∞ 

1 2iπ

an n −s

n=1



b+i∞

(w)(nδ)−w dw =:

b−i∞

∞   n=1

b+i∞

f n (w)dw.

b−i∞

But, by (7.4.3), we see that ∞   n=1

b+i∞

| f n (w)|dw ≤

b−i∞

∞ 

|an |n −σ Cb

n=1

= Cb

∞ 

1 2π



∞ −∞

e−ρ|y| (nδ)−b dy

|an |n −σ−b < ∞ since σ + b > σa ( f ).

n=1

Therefore, we can permute sums and integrals to get the claimed formula for gδ . We will now push this integral representation to the left, and show that, for s ∈ K , we have the crucial approximation formula gδ (s) − f (s) =

1 2π



∞

−∞

(−c + i y) f (s − c + i y)δ c−i y dy

(7.4.10)

where 0 < c < 1 is such that σ0 − c > α. To that effect, we apply the Cauchy formula to the function ϕ(w) = (w) f (s + w)δ −w , which verifies Res(ϕ, 0) = f (s), along the rectangle with vertices b ± i T , −c ± i T . We get 

b+i T

2iπ f (s) =

 ϕ(w)dw+

b−i T



+

−c+i T

 ϕ(w)dw+

−c−i T

ϕ(w)dw =:

4 

ϕ(w)dw

−c+i T

b+i T

b−i T

−c−i T

I j (T ).

j=1

We claim that, as T → ∞, we have for some positive constants C, λ:  |I2 (T )| ≤ C

b −c

e−λT (A + T B )δ −b d x = O(T B e−λT ) = o(1).

Indeed, the following holds.

7.4 A Generalized and Uniform Carlson Formula

223

Fig. 7.2 Backward shift

w ∈ [−c + i T, b + i T ] and s ∈ K =⇒ Re(s + w) ≥ σ0 − c > α, f is of finite order in Cσ0 −c and we have for  the uniform horizontal estimate (7.4.4), with arg(x + i T ) ≥ arctan(T /b) for −c ≤ x ≤ b. Similarly, I4 (T ) → 0, while I1 (T ) → gδ (s) and I3 (T ) → −I where I is 2iπ times the RHS of (7.4.10), which is therefore proved, and actually holds for any s ∈ Cα with Re s ≥ σ0 . As a consequence of (7.4.10), we shall see that (see Fig. 7.2 above) sup s∈K

1 T



T

(|gδ (s + it) − f (s + it)|2 dt ≤ Cδ 2c

(7.4.11)

0

where the constant C only depends on K and c. Indeed, making a vertical translation s → s + it, which verifies Re(s + it) = Re s ≥ σ0 , (7.4.10) and (7.4.3) give, for 0 ≤ t ≤ T and some constant C:  ∞ e−ρ|y| | f (s − c + i(t + y)|δ c dy. |gδ (s + it) − f (s + it)| ≤ C −∞

224

7 Voronin-Type Theorems

We now use Cauchy-Schwarz with respect to the measure e−ρ|y| dy and split the new integral in two parts to obtain  |gδ (s + it) − f (s + it)|2  δ 2c   2c −ρT e + δ

2T

e

∞

−∞

−ρ|y|

−2T

e−ρ|y| | f (s − c + i(t + y))|2 dy 

| f (s − c + i(t + y))| dy . 2

(7.4.12)

Here, we used the fact that, due to the finite order assumption on f , and with uniform estimates for s ∈ K , one has:

 −ρ|y| | f (s − c + i(t + y))|2 dy  |y|>2T e−ρ|y| A2 + |t|2B + |y|2B dy |y|>2T e  T 2B |y|>2T e−ρ|y| dy + |y|>2T |y|2B e−ρ|y| dy  e−ρT . We now integrate (7.4.12) with respect to t ∈ [0, T ] and use Fubini as well as our L 2 -assumption, which did not yet intervene. We get, in a uniform manner on K ,   δ

2c

Te

−ρT

 +

T

|gδ (s + it) − f (s + it) |2 dt

0

2T

e −2T

−ρ|y|



T

 | f (s − c + i(t + y))| dt dy  δ 2c T. 2

0

Here, we used the fact that, for |y| ≤ 2T , 

T



y+T

| f (s − c + i(t + y))| dt = 2

0



| f (s − c + iv)|2 dv

y

=

y+T



y

···−

0

···  T

0

due to our L 2 -assumption, to the membership of Re(s − c) in a compact subset of (α, β), and to the inequality |y| ≤ 2T . This proves (7.4.11). The rest will be routine. It will be convenient to set (h denoting an arbitrary function) 

1/2  1 T 2 |h(t)| dt , f s (t) = f (s + it), gδ,s (t) = gδ (s + it), MT (h) = T 0 ∞ ∞ 1/2 1/2   2 −2nδ −2Re s (s) = |a | e n , Σ(s) = |an |2 n −2Re s . Σδ n n=1

n=1

Recalling that σ0 = inf s∈K Re s > α, we first claim that

7.4 A Generalized and Uniform Carlson Formula ∞ 

225

|an |2 n −2σ0 < ∞.

(7.4.13)

n=1

Indeed, using (7.4.11) and the triangle inequality, we have for s ∈ K MT (gδ,s ) ≤ MT (gδ,s − f s ) + MT ( f s )  δ c + 1  1. Passing to the limit on T , we get: Σ δ (s)  1. Passing to the limit on δ and using Fatou, we get Σ(s)  1 as well, and this gives (7.4.13). Secondly, we claim that, for fixed δ > 0, we have



lim sup MT (gδ,s ) − (s) = 0. δ

T →∞ s∈K

(7.4.14)

A brute force calculation actually gives (since |an | ≤ Cδ enδ/2 ):  1 sup |MT2 (gδ,s ) − Σδ2 (s)| ≤ T2 |am | |an |(mn)−σ0 e−(m+n)δ | log(m/n)| s∈K =n  m−(m+n)δ/2 1 ≤ CTδ e log(1+1/n) m>n    C 1 ≤ CTδ e−nδ/2 log(1+1/n) e−mδ/2 =: Tδ , n≥1

m≥1

√ proving (7.4.14) since the function x → x is uniformly continuous on R+ . Finally,



let ε > 0. Using MT ( f s ) − MT (gδ,s ) ≤ MT ( f s − gδ,s ), we write sup |MT ( f s ) − Σ(s)| s∈K



≤ sup MT ( f s − gδ,s )+ sup MT (gδ,s ) − Σδ (s)| + sup Σδ (s) − Σ(s)| s∈K

s∈K

s∈K

≤ 2ε + sup |MT (gδ,s ) − Σδ (s)| s∈K

by adjusting δ, thanks to (7.4.11) and to (7.4.13). Now, using (7.4.14), we get sup |MT ( f s ) − Σ(s)| ≤ 3ε for all T ≥ Tε . s∈K

Since the function x → x 2 is uniformly continuous on bounded subsets of R+ , this ends the proof of Theorem 7.4.4. 

226

7 Voronin-Type Theorems

Remark. Carlson’s formula should be considered as a delicate fact. In particular,  −s a n ∈H∞ and f (it) = limσ→0+ f (σ + it), which exists almost let f (s) = ∞ n=1 n everywhere, since we have f ∈ H ∞ (C0 ). By Carlson’s formula, we know that lim

T →∞

1 T



T

| f (σ + it)|2 dt =

0

∞ 

|an |2 n −2σ , σ > 0.

(7.4.15)

n=1

Therefore, we might be tempted to put σ = 0 in the above, but this is wrong! Wrong statement: Equation (7.4.15) holds for σ = 0 and for f (it). A nice counterexample to that statement was given in [24]. −s Another related pathology is the following: ∞ n=1 an n , the Dirichlet series of an H∞ -function f , can diverge everywhere on the line Re s = 0, even if f is continuous on the closed right ([25]). To get convergence, we have to shift by 1/2: the half-plane −1/2−it a n converges for almost every t ([26] and also [27]), Dirichlet series ∞ n=1 n as we proved in Chap. 6. And this is optimal ([25]).

7.5 Joint Universality of the Singleton λ = (L(s, χ j )) 7.5.1 Notations and the Idea of Proof of Theorem 1.2 In the beginning of this section, it is convenient to recall some notations. We always denote by K and U a compact set and a Jordan domain such that K ⊂ U ⊂ U ⊂ , σ0 := inf Re s > s∈U

1 2

(7.5.1)

and by Y = (C(K ))r and H = (B 2 (U ))r the associated vector spaces as in (7.2.6), equipped with the norms (7.2.7). We will fix once and for all an r -tuple f = ( f 1 , . . . , fr ) ∈ (H ∗ ())r . For any g = (g1 , . . . , gr ) ∈ H , we have (see 7.2.7):  2 |g|2 d A. g H = U

For χ1 , . . . , χr distinct characters mod q, we have set     L j (s) = L s, χ j , λ = L 1 , . . . , L r and j

L N (s) = L N (χ j , s) =

N  k=1

(1−χ j ( pk ) pk−s )−1 , λ N =(L 1N , . . . , L rN ).

7.5 Joint Universality of the Singleton λ = (L(s, χ j ))

227

For z = (z k ) ∈ T∞ , we have also set j

L N (s, z) =

N 

(1−z k χ j ( pk ) pk−s )−1 ,

(7.5.2)

k=1

  λzN = L 1N (., z), ..., L rN (., z) . With the notations of Chap. 2, μ is the Haar measure of T∞ , and e = (1, ..., 1, ...),

  K t e = p1−it , ..., pk−it , ... ∈ T∞

where K t is the Kronecker flow, not to be confused with the translation operators Tt of the first section. We fix ε > 0. The sets V and D will be as in (7.3.6) and (7.3.7), respectively, with δ = ε. We will use the notation g ∼ h to indicate that g − hY ≤ ε. We wish to show that, for many positive t’s, f is close to Tt λ. This will be done in three steps: (1) Let N0 be a positive integer. We have for some integer N ≥ N0 ( f ∼ Tt λ N for t ∈ D, where dens(D) > 0.

(7.5.3)

The proof uses the Kronecker approximation theorem. (2) We have for all n ≥ N Tt λ N ∼ Tt λn for t ∈ An ⊂ D, where dens(An ) ≥

2 dens(D). 3

(7.5.4)

The proof uses the Birkhoff–Khinchin theorem. Note that An ⊂ D. (3) We have for some n ≥ N 1 Tt λn ∼ Tt λ for t ∈ Bn , where dens(Bn ) ≥ 1 − dens(D). 3

(7.5.5)

The proof uses the Carlson identity. The three steps put together give the result. Indeed, fix n ≥ N verifying (7.5.4) and (7.5.5) and set  = An ∩ Bn ⊂ D. Then we have (see the exercises) dens() ≥ dens(An ) + dens(Bn ) − 1 ≥ (1/3)dens(D) > 0 and moreover the triangle inequality gives, for t ∈ ,  f − Tt λY ≤  f − Tt λ N Y + Tt λ N − Tt λn Y + Tt λ N − Tt λY ≤ 3ε.

228

7 Voronin-Type Theorems

7.5.2 Details of Proof Let us now detail the three steps. We will make repeated use of the following simple technical lemma. Lemma 7.5.1. Let ε > 0 and 0 < c < 1. Then, we can find η > 0 such that, for any set D ⊂ R+ of positive density and any g ∈ H , the following implication holds : 1 lim sup T →∞ T

 0

T

1 D (t)Tt g2H dt ≤ η 2 dens(D)

=⇒dens{t ∈ D; Tt gY ≤ ε} ≥ (1 − c)dens(D). Proof. Let C be the constant appearing in (7.2.8). Let η 2 = cε2 /C 2 , let m be the Lebesgue measure on R and X be the function defined by X (t) = Tt g H . The Markov–Chebyshev inequality reads m{t ∈ D ∩ [0, T ] ; X (t) > ε/C} ≤

C2 ε2



T

1 D (t) X 2 (t) dt. 0

Dividing by T , letting T tend to infinity, using the assumption and the inequality Tt gY ≤ C X (t), we get dens{t ∈ D; Tt gY > ε} ≤ dens{t ∈ D; X (t) > ε/C} ≤

C 2 η2 dens(D) ε2

= c dens(D).

Passing to the complement, we get the conclusion.

7.5.2.1



Proof of Step One

The required lemma is nothing but Theorem 7.3.2, which we recall here under the form of a lemma. Lemma 7.5.2. Let ε > 0, 0 < c < 1 and N0 a positive integer. Then, we can find D ⊂ R+ , of positive density, and N ≥ N0 such that t ∈ D =⇒  f − Tt λ N Y ≤ ε.

7.5.2.2

(7.5.6)

Proof of Step Two

The key for that second step will be the following lemma, in which D and N0 are as in Lemma 7.5.2. An important feature of that lemma is that the set An obtained will lie inside the previous favourable set D. This also explains why we cannot use the tempting Carlson identity for that step, as we shall do for the next one.

7.5 Joint Universality of the Singleton λ = (L(s, χ j ))

229

Lemma 7.5.3. In Lemma 7.5.2, we can choose D, N0 and then N ≥ N0 so as to have, for each n > N , dens(An ) := dens{t ∈ D; Tt λn − Tt λ N Y ≤ ε}

(7.5.7)

≥ (2/3)dens(D). Proof. Let g = λn − λ N , n > N ≥ N0 . We will check by a long computation the assumptions of Lemma 7.5.1 for g, ε, c = 1/3 and the corresponding η. We first have by Fubini and (7.3.7):  1 T 1 D (t)Tt λn − Tt λ N 2H dt = T 0    1 T 1V (K t e) |Tt λn (s) − Tt λ N (s)|2 d A(s) dt T 0 U    T 1 1V (K t e) |λn (s, K t e) − λ N (s, K t e)|2 d A(s) dt = T 0 U     T 1 2 1V (K t e)|λn (s, K t e) − λ N (s, K t e)| dt d A(s). = U T 0 Now, the Birkhoff–Khinchin and Lebesgue theorems give  1 T 1 D (t)Tt λn − Tt λ N 2H dt I := lim sup T T →∞ 0    2 1V (z)|λn (s, z) − λ N (s, z)| dμ(z) d A(s). = U

T∞

By definition, if z ∈ V and s ∈ U , we have √ |λ N (s, z)| ≤ sup | f (s)| + ε r ≤ M, s∈U

where M < ∞ only depends on f . This will allow us, in the estimation of I , to perform two independent integrations, in the sense of probabilists. Indeed, we can write I =

r    j=1

U

 T∞

1V (z)|λnj (s, z)

−

j λ N (s, z)|2 dμ(z)

d A(s)

! j

λn (s, z)

2

≤M 1V (z) 1 − j

dμ(z) d A(s) λ N (s, z) T∞ j=1 U !  r  j  λn (s, z)

2

2 = M μ(V )

1 − j

dμ(z) d A(s) λ N (s, z) T∞ j=1 U 2

r  



230

7 Voronin-Type Theorems

since V only depends on z 1 , . . . , z N and

j

λn (s,z) j λ N (s,z)

only on z N +1 , . . . , z n . Recall that

σ0 = inf s∈U Re s > 1/2 and set σ = Re s. The integral K j (s) between square brackets is estimated by brute force:  K j (s) =  =

≤

Tn−N

Tn−N



n

2 

(1 − χ j ( pk )z k pk−s )−1 dz N +1 · · · dz n

1 −



k=N +1 n 

2

χ j (αk pk )z kαk pk−sαk dz N +1 · · · dz n

α N +1 ,...,αn ≥0, k=N +1 max αk >0 n  

 α N +1 ,...,αn ≥0, max αk >0

−2σ

k=N +1

α pk k



≤

l>N0

l −2σ ≤

N01−2σ0 . 2σ0 − 1

Here,  we used Parseval’s formula, and the fact that all the implied integers l = nk=N +1 pkαk are ≥ p N +1 > N0 . Now, coming back to the integral I , we see that, since μ(V ) = dens(D): I ≤ (M 2 r ) × dens(D) × N

N01−2σ0 . 2σ0 − 1

1−2σ0

Adjust N0 so big that M 2 r 2σ0 0 −1 ≤ η 2 . We are then in a position to apply Lemma 7.5.1 to get the conclusion. 

7.5.2.3

Proof of Step Three

The final lemma needed is the following. Lemma 7.5.4. If D, N0 and N ≥ N0 are as in Lemma 7.5.3, we have for some n>N: dens{t ∈ R+ ; Tt λn − Tt λY ≤ ε} ≥ 1 − (1/3)dens(D). Proof. Recall that P + (k) is the largest prime divisor of k. We have L j (s) − L nj (s) =

∞ 

b(k, n, j)k −s

k=1

with



b(k, n, j) = χ j (n) if P + (k) > pn b(k, n, j) = 0 if P + (k) ≤ pn .

(7.5.8)

7.5 Joint Universality of the Singleton λ = (L(s, χ j ))

231

We will apply Lemma 7.5.1 with the set R+ , ε > 0, c = (1/3) dens(D), where D is the set of Lemma 7.5.2, and the corresponding η = η(ε, c). Set δ 2 = η 2 /r A(U ) where A(U ) is the area measure of U . Now, if χ j is not the principal character mod j q, the functions L j , L n verify the assumptions of Carlson’s Theorem 7.4.4 with, e.g.. α = 1/2 and β = 1, as we saw in Theorems 7.1.1 and 7.1.2. If it is principal, we consider instead the alternate Dirichlet series R(s) = (1 − 21−s )(L j (s) − L nj (s)) =

∞ 

(−1)k−1 b(k, n, j)k −s .

k=1

The zero of 1 − 21−s compensates the pole of L j at 1, therefore, R is holomorphic in C1/2 . Moreover, the factor 1 − 21−s is bounded above and below on any substrip {s ∈ ; 1/2 < u < Re s < v < 1}, therefore we can apply Carlson’s formula to R, then get rid of 1 − 21−s and (−1)k−1 and in any case we will be able to claim the following. Given ε > 0, we can find Tε such that, for all 1 ≤ j ≤ r and all s ∈ U , T ≥ Tε implies 1 T



T

|L (s + it) − j

0

≤

L nj (s

∞ δ2  + it)| dt ≤ + |b(k, n, j)|2 k −2Re s 2 k=1 2

∞ ∞   δ2 δ2 + + k −2Re s ≤ k −2σ0 ≤ δ 2 2 2 k> p k> p n

n

(by adjusting n). By adding those inequalities, we get if T ≥ Tε and s ∈ U, then

1 T



T

|λ(s + it) − λn (s + it)|2 dt ≤ r δ 2 .

0

Integrating over U with respect to d A(s) and permuting, we obtain lisup

T →∞

1 T



T 0

Tt λ − Tt λn 2H dt ≤ r δ 2 A(U ) = η 2 .

Lemma 7.5.1 now gives the conclusion.



Concluding remarks: Recall that  denotes the critical half-strip {s; 1/2 < Re s < 1}. The initial statement of Voronin (1975) was stated for the uniform approximation in some disc and did not concern the whole of  (see [28]). A significant improvement was obtained independently by Bagchi ([10]) and Reich ([29]), under the following local form, in which A(K ) denotes the algebra of functions which are both continuous on the compact set K and analytic in its interior.

232

7 Voronin-Type Theorems

Theorem 7.5.5. Let K be a compact subset of  with connected complement and f ∈ A(K ) without zeros. Then, for any ε > 0, there exists a set D ⊂ R+ of positive lower density such that t ∈ D =⇒ sup | f (s) − ζ(s + it)| ≤ ε. s∈K

The reader should be able to reconstruct a proof from a careful reading of this chapter, once he has noticed that if f ∈ A(K ) has no zeros, it can be written as f = eg with g ∈ A(K ). A slight improvement has been obtained on this point by Gauthier ([30]): if one only assumes that f has no zeros on the interior of K, one has the same conclusion. The topological restriction on K is due to the use in the proof of Mergelyan’s approximation theorem. A similar local form was obtained by Bagchi ([10]) for the joint approximation of an r -tuple of non-vanishing functions of A(K ) by a translate of an r -tuple of L-functions. Another very striking statement he obtained, which definitely connects the Riemann hypothesis and universality problems for vertical translates, is the following [10]. Theorem 7.5.6. The Riemann hypothesis holds if and only if, for every compact subset K of the critical half-strip  with connected complement and every ε > 0, there exists a set D ⊂ R+ of positive lower density such that t ∈ D =⇒ sup |ζ(s) − ζ(s + it)| ≤ ε. s∈K

Following those seminal results on universality, much work has been devoted to various extensions, for a detailed exposition of which we refer to the Lecture Notes of Steuding ([31]). For example, the zeta function can be replaced, under certain conditions on the parameters, by Lerch functions L(s, λ, a) =

∞  exp(2iπλn) n=0

(n + a)s

which themselves extend Hurwitz functions since L(s, 0, a) = ζ(s, a). Another beautiful recent result due to Gauthier ([32]), and free from a priori assumptions on the zeros of the function, is as follows. Theorem 7.5.7. If f ∈ H (), there exists an increasing sequence (K n ) of compact sets of , whose union is , such that, for each n and each ε > 0, there exists a set D = D(n, ε) ⊂ R+ with positive lower density such that t ∈ D =⇒ sup | f (s) − ζ(s + it)| ≤ ε. s∈K n

7.5 Joint Universality of the Singleton λ = (L(s, χ j ))

233

In this chapter, we did not try to be either exhaustive or up-to-date, but only to give a complete presentation (as far as possible) of the results of Bagchi and Voronin, to encourage the reader to study more recent or more advanced work on this fascinating, and still quite active, subject.

7.6 Exercises 1. Let H be a separable Hilbert space, finite-dimensional or not. Give a simple example of a sequence (xn ) of vectors of H verifying the assumptions of Theorem 7.2.1  p ||x and even ∞ n || < ∞ for each p > 1. n=1 2. Let q, b be two coprime integers, and λ(s) =



n −s .

n≡b mod q

Show that the function λ is universal. 3. Let A, B ⊂ R+ . Detail the proof of the inequality dens(A ∩ B) ≥ dens(A) + dens(B) − 1. Give examples where the equality is strict, or attained. 4. (A counterexample). Let U = D\[1/2, 1[, a bounded domain of C whose boundary is not a Jordan curve, although it is star-shaped with respect to the origin. (i) Why is a Jordan domain a Carathéodory domain? (ii) Show that U is not a Carathéodory domain. (iii) Show that, for α > −1 and α not an integer, the function f (z) = (z − 1/2)α is in the Bergman space B 2 (U ). (iv) Show that f is not in the closure of polynomials in B 2 (U ). (Hint: if a sequence (Pn ) of polynomials converges to f in B 2 (U ), it converges locally uniformly on D.) 5. (i) Let a0 , ..., a M ∈ C with a0 = 0. Show that there exists an entire function f , without zeros, such that D k f (0) = ak for 0 ≤ k ≤ M, where D denotes the derivative: D f = f  . (Hint: look for a function f of the form f = e P where P is a polynomial.) (ii) Let χ1 , . . . , χr be r distinct characters mod q, and for 1 ≤ j ≤ r , we write L j = L(., χ j ). Let N be a non-negative integer. Fix a real number σ such that 1/2 < σ < 1. Show that the map T : R → Cr (N +1) defined by T (t) = {D k L j (σ + it)}0≤k≤N , 1≤ j≤r

has dense range.

234

7 Voronin-Type Theorems

(iii) Using the previous question, show that if the continuous (on C N +1 ) functions Fk, j , 0 ≤ k ≤ N , 1 ≤ j ≤ r are such that 

s j+k(N +1) F j,k (L j (s), DL j (s), . . . , D N L j (s)) = 0

0≤k≤N , 1≤ j≤r

for all s ∈ C\{1}, then F j,k = 0 for all j, k. In other terms, there is no non-trivial algebraic-differential equation relating L 1 , . . . , L r . 6. Let 0 < a ≤ 1 and consider the Hurwitz zeta function ζ(s, a) =

∞ 

(n + a)−s .

n=0

Is this function universal? Show that this is the case if a is a transcendental number. The result is due to Bagchi  ([33]) and−sGonek ([34]) independently. 7. Let again ζ(s, a) = ∞ be the Hurwitz zeta function, where a is a n=0 (n + a) fixed transcendental number, 0 < a ≤ 1. (i) Show that the numbers log(n + a), n ≥ 0 are rationally independent. (ii) Let δ > 0. Fix an integer N such that N  n=0

(n + a)−1−δ >

∞ 

(n + a)−1−δ

n=N +1

and let α(n) = 1 for n ≤ N and α(n) = −1 for n > N . Show that Z (s) :=  ∞ −s has a zero σ1 in (1, 1 + δ). n=0 α(n)(n + a) (iii) Show that there exists a number η1 > 0 such that σ1 + η1 < 1 + δ; σ1 − η1 > 1; ε =

min |Z (s)| > 0.

|s−σ1 |=η1

(iv) Using the Kronecker theorem, show that there exists a real number τ such that Re s ≥ σ1 − η1 =⇒ |ζ(s + iτ , a) − Z (s)| < ε. (v) Using Rouché’s theorem, show that ζ(s, a) has a zero in the vertical strip {s; 1 < Re s < 1 + δ}, and therefore has infinitely many zeros. The result also holds for the case a rational and a = 1, a = 1/2 and is due to Davenport and Heilbronn ([35]). The more difficult (but still positive) result for a algebraic irrational is due to Cassels [36].

References

235

References 1. H. Davenport, Multiplicative Number Theory, 3rd edn. (Springer, 1980) 2. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres (Institut Elie Cartan, 1995) 3. E. Titchmarsh, The Theory of Functions (Clarendon Press, 1932) 4. F. Bayart, E. Matheron, Dynamics of Linear Operators (Cambridge University Press, 2007) 5. A. Ivic, The Riemann Zeta Function, Theory and Applications (Dover Publications, 1985) 6. O. Ramaré, Private Communication (2012) 7. E. Landau, Handbuch des Lehre von der Verteilung der Primzahlen, 2nd ed. (Chelsea University Company, 1953) 8. H. Bohr, Zur Theorie der Riemannschen Zeta funktion im kritischen Streife. Acta Math. 40, 67–100A (1915) 9. S. Jaffard, The spectrum of singularities of Riemann’s function. Rev. Math. Iberoamericana 12(2), 441–460 (1996) 10. B. Bagchi, A joint universality theorem for Dirichlet L-functions. Math. Z. 181, 319–334 (1982) 11. V. Drobot, Rearrangements of series of functions. Trans. Amer. Math. Soc. 142, 239–248 (1969) 12. E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme. J. Reine Angew. Math. 143 (1913) 13. J.P. Kahane, Sur les polynômes à coefficients unimodulaires. Bull. London. Math. Soc. 12, 321–342 (1980) 14. H. Queffélec, B. Saffari, On Bernstein’s inequality and Kahane’s ultraflat polynomials. J. Fourier Anal. Appl. 2, 519–582 (1996) 15. Ch. Pommerenke, Boundary Behaviour of Conformal maps (Springer, 2010) 16. W. Rudin, Real and complex analysis, 3rd edn. (1984) 17. G. Szegö, Conformal mapping of the interior of an ellipse onto a circle. Amer. Math. Monthly 356(3), 881–893 (2004) 18. P. Bourdon, Density of the polynomials in the Bergman space. Pacific J. Math. 130, 215–221 (1987) 19. A. I. Markushevich, Theory of Functions of a Complex Variable (Chelsea, 1977) 20. G.G. Lorentz, Approximation of functions, 2nd edn. (AMS Chelsea Publishing, 1986) 21. H. Montgomery, B. Vaughan, Multiplicative number theory I (Cambridge University Press, 2007) 22. P. Samuel, Théorie algébrique des nombres (Hermann, 1967) 23. E. Andrews, R. Askey, R. Roy, Special Functions, vol. 71. (Cambridge University Press, 1999) 24. E. Saksman, K. Seip, Integral means and boundary limits of Dirichlet series. Bull. Lond. Math. Soc. 41(3), 411–422 (2009) 25. F. Bayart, S. Konyagin, H. Queffélec, Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series. Real Anal. Exchange 29(2), 557–586 (2003–2004) 26. H. Hedenmalm, E. Saksman, Carleson’s convergence theorem for Dirichlet series. Pacific. J. Math. 208, 85–109 (2003) 27. S. Konyagin, H. Queffélec, The translation 1/2 in the theory of Dirichlet series. Real Anal. Exchange 27, 155–175 (2001–2002) 28. S. M. Voronin, Theorem on the ‘universality’ of the Riemann zeta-function, Izv. Akad. Nauk SSSR, Ser. Matem. 39, 475–486 (Russian). Mat. USSr. Izv. 9, 443–445 (1975) 29. A. Reich, Universelle Wertverteilung von Eulerprodukten, Nach. Akad. Wiss. Göttingen, Math.Physik. Kl., 1–17 (1977) 30. P. Gauthier, Approximating all meromorphic functions by linear motions of the Riemann zetafunction. Comp. Methods Funct. Theory 12(2), 517–526 (2012) 31. J. Steuding, Value-Distribution of L-functions, 2nd edn. (Springer Lecture Notes 1877, 2007) 32. P. Gauthier, Approximation of functions having zeros by translates of the Riemann zetafunction, preprint

236

7 Voronin-Type Theorems

33. B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet series. Ph.D. Thesis, Indian Statistical Institute, Calcutta, 1981 34. S. M. Gonek, Analytic properties of zeta and L-functions. Ph.D. Thesis, University of Michigan, 1979 35. H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series. Journal Lond. Math. Soc. 11, 181–185 (1931) 36. J. Cassels, Footnote to a note of Davenport and Heilbronn. Journal Lond. Math. Soc. 36, 177–184 (1961)

Chapter 8

Composition Operators on the Space H2 of Dirichlet Series

8.1 Introduction The general framework for composition operators acting on a Banach space X of functions analytic on a domain Ω of Cd , 1 ≤ d ≤ ∞, is the following: we always assume that X is continuously embedded in the Fréchet space H (Ω) := Hol (Ω), so that the point evaluations δa , δa ( f ) = f (a) are continuous linear forms on X for each a ∈ Ω. If now ϕ is an analytic self-map of Ω, the composition operator Cϕ : X → H (Ω) is defined by Cϕ ( f ) = f ◦ ϕ and all the game consists first in deciding whether Cϕ : X → X (in that case, Cϕ is bounded by the closed graph theorem, and ϕ will be called a symbol, or an admissible symbol) and then in comparing the operator-theoretic properties of Cϕ and the function-theoretic properties of ϕ. In case Cϕ : X → X , a formal but useful property, in which Cϕ∗ designates the transpose of Cϕ , is: (Cϕ )∗ (δa ) = δϕ(a)

for all a ∈ Ω.

(8.1.1)

When X is a Hilbert space, the evaluation δa is given (via Riesz’s theorem) by the scalar product with an element K a of X , called the reproducing kernel of X at a, that is δa ( f ) =  f, K a , and the Eq. (8.1.1) takes the form (here Cϕ∗ denotes the adjoint of Cϕ ): (8.1.2) (Cϕ )∗ (K a ) = K ϕ(a) for all a ∈ Ω. The relations (8.1.1) or (8.1.2) are called the mapping equations and turn out to be very useful.

© Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2_8

237

238

8 Composition Operators on the Space H2 of Dirichlet Series

The real interest in those operators started indeed (in the case of the Hardy space X = H 2 of the unit disc Ω = D, more generally of H p ) with the celebrated Littlewood’s subordination principle [1], which tells the following. Theorem 8.1.1 Let ϕ : D → D be analytic and let 1 ≤ p ≤ ∞. Then, the composition operator Cϕ formally defined by Cϕ ( f ) = f ◦ ϕ is a continuous operator: H p → H p , and more precisely: (1 − |ϕ(0)|2 )−1/ p ≤ Cϕ ≤

 1 + |ϕ(0)| 1/ p 1 − |ϕ(0)|

.

In particular, Cϕ is a contraction if and only if ϕ(0) = 0. Observe that the analyticity of f ◦ ϕ, the composition of two analytic functions, is obvious. The membership of f ◦ ϕ in H 2 is not, and this is the core of Theorem 8.1.1, even for simple symbols as ϕ(z) = z +2 1 (see Exercise 1). Much after Littlewood’s paper, composition operators on various Hilbert spaces of analytic functions on D (Hardy, Bergman, Dirichlet, etc.) have been intensively studied (compactness, membership in Schatten classes, decay of singular numbers, etc.), in particular, by the first named author (see, e.g. [2–6]); the infinite-dimensional case was also touched (see [7]). This study of holomorphic functions, in high or even infinite dimension, is also performed in depth in the recent book [8]. In the case of the Hardy space H2 of Dirichlet series living on the half-plane C1/2 , and of an analytic self-map ϕ of C1/2 , the situation is quite different. If it is still obvious that f ◦ ϕ is analytic on C 1/2 , not any analytic function on this half-plane lets itself expand under the form bn n −s and whereas Theorem 8.1.1 is impressive by its extreme generality, here we will have to make strong restrictions on the “symbol” ϕ. In a seminal paper [10], Gordon and Hedenmalm were able to give a full description of those admissible symbols. We will detail their work in this chapter. Let us mention that plenty of fascinating questions (characterization of compactness, of the admissible symbols for H p when p = 2, etc.) remain open. For our purpose, we will have to introduce two classes of functions. We stick to the notations of the previous chapters, in particular, D denotes the space of convergent Dirichlet series, namely, ψ(s) =

∞ 

cn n −s

with σc (ψ) < ∞.

n=1

Now, our two classes are 1. The class D0 of analytic functions ϕ : C1/2 → C1/2 which can be written as ϕ(s) = c0 s + ψ(s) where c0 ∈ N ∪ {0} and ψ ∈ D.

(8.1.3)

8.1 Introduction

239

We will occasionally consider, for θ, ν ∈ R, the class Dθ,ν of analytic functions ϕ : Cθ → Cν which can be written as in (8.1.3), even if the main case will remain θ = ν = 1/2. Note that D0 = D1/2, 1/2 . 2. The class G of analytic functions ϕ(s) = c0 s + ψ(s) which satisfy (i) ϕ ∈ D0 . (ii) ψ has an analytic continuation in C0 and moreover σu (ψ) ≤ 0. (iii) ψ(C0 ) ⊂ C0 if c0 ≥ 1 and ψ(C0 ) ⊂ C1/2 if c0 = 0. The integer c0 is called the characteristic of the symbol ϕ. When c0 = 0, we are again in the class D0 . The case c0 ≥ 1 should be interpreted as ϕ(∞) = ∞. Indeed, the symbols with positive characteristic are nicer than the remaining ones, quite the same as analytic self-maps of D behave better (see Theorem 8.1.1) when they fix the origin. In most proofs to come, we will have to treat separately the cases c0 ≥ 1 and c0 = 0, the latter case inducing severe additional difficulties. This will be made more explicit in the sequel, and extensions of [10], as well as open problems, will be indicated. The class G was coined in [10]. We called it the Gordon– Hedenmalm class in [9] (even though the authors of [10] did not use that terminology (!)), with a slight improvement to be described in next section. This class is rather restricted (indicating that few analytic maps ϕ : C1/2 → C1/2 induce composition operators Cϕ : H2 → H2 ), and its definition involves analyticity and mapping conditions which may seem artificial at first glance, but describe exactly the symbols generating bounded composition operators Cϕ : H2 → H2 .

8.2 The Main Theorem The main theorem of this chapter is the following (a more precise form will be given later): Theorem 8.2.1 Let ϕ : C1/2 → C1/2 , analytic. The following are equivalent: 1. Cϕ : H2 → H2 . 2. ϕ ∈ G. The proof is fairly involved, and will be conveniently divided into two theorems. We will follow Gordon and Hedenmalm rather closely (providing some more details and comments, and sometimes adopting a slightly different point of view).

8 Composition Operators on the Space H2 of Dirichlet Series

240

8.3 The Arithmetic Theorem In this section, we prove the following, mainly arithmetic, result: Theorem 8.3.1 Let ϕ : C1/2 → C1/2 , analytic. The following are equivalent: 1. Cϕ : H2 → D. 2. ϕ ∈ D0 . Proof We will rely on two lemmas.  −s Lemma 8.3.1 Let h(s) = ∞ ∈ D. Then n=r an n lim r s h(s) = ar .

s→∞

(8.3.1)

Indeed, fix σ0 > σa (h). For s ≥ σ0 , we write r s h(s) = ar +



an (r/n)s with |an (r/n)s | ≤ |an (r/n)σ0 | =: |bn |.

n>r



We have |bn | < ∞ and moreover lims→∞ an (r/n)s = 0 for each n > r . The Weierstrass M-test now gives the result. Lemma 8.3.2 Let c be a real number such that n c is an integer for all n ∈ N. Then, c is a non-negative integer. Indeed, let f : [1, ∞[→ C be a C ∞ -function. We consider the k-iterated difference (k f )(n), inductively defined as 0 f (n) = f (n) and (k+1 f )(n) = (k g)(n) where g(n) = f (n + 1) − f (n) or again (k f )(n) =

   k  k f (n + j) = (−1)k− j f (k) (n + t1 + · · · + tk )dt1 · · · dtk . k j [0,1] j=0

Suppose now that c is not an integer. Let k be an integer > c, and let us take here f (t) = t c , with f (k) (t) = c(c − 1) · · · (c − k + 1)t c−k . Since k > c, the integral expression shows that limn→∞ (k f )(n) = 0. Since c is not an integer, f (k) has constant sign, and the integral expression again shows that the integer (k f )(n) = 0 and hence |(k f )(n)| ≥ 1. This is a contradiction! Let us now prove the necessity of the condition ϕ ∈ D0 in Theorem 8.3.1. By hypothesis, we can write k −ϕ(s) =

∞  n=N (k)

bn(k) n −s ∈ D

(8.3.2)

8.3 The Arithmetic Theorem

241

dk with b(k) N (k) =: ck  = 0. We set ck = e . Lemma 8.3.1 now shows that

(N (k))s k −ϕ(s) = exp(s log N (k) − ϕ(s) log k) → ck as s → ∞.

(8.3.3)

Let θ > σc (k −ϕ ) and let f (s) = s log N (k) − ϕ(s) log k, continuous in the halfplane Cθ . Thanks to (8.3.3), we can adjust θ large enough to get f (Cθ ) ⊂



 U (k) + 2iπl =: Ol , l∈Z

l∈Z

where U (k) is a small open neighbourhood of dk , so that the Ol are disjoint. But f (Cθ ) is connected, so that f (Cθ ) ⊂ Oq for a single q ∈ Z, and more precisely s log N (k) − ϕ(s) log k → dk + 2iqπ as s → ∞.

(8.3.4)

Dividing by s log k, for k > 1, we obtain lim

s→∞

log N (k) ϕ(s) = =: c0 . s log k

In particular, the real number c0 does not depend on k, and moreover k c0 = N (k) is an integer for all k ∈ N. By Lemma 8.3.2, we see that c0 is a non-negative integer. Let ψ(s) = ϕ(s) − c0 s. We now show that ψ ∈ D. Multiplying (8.3.2) by k c0 s , we obtain k −ψ(s) =

∞  n=k c0

bn(k)

n −s . k c0

We drop the superscript and write this relation as



1 −s 2 −s + B2 1 + c + · · · =: B0 + h(s). k −ψ(s) = B0 + B1 1 + c 0 k k0

(8.3.5)

Here, the notation B j stands for the shifted coefficients bk(k) c0 + j . Also observe that, by Lemma 8.3.1 applied to  bn(k) n −s , n>k c0

it holds lims→∞ |h(s)| = 0. Next, combining (8.3.5) with (8.3.4), we obtain for s large enough to ensure that |h(s)| < |B0 |: −ψ(s) log k = log(1 + h(s)/B0 ) + log B0 + 2iπq, where log is the principal branch of the logarithm, namely,

8 Composition Operators on the Space H2 of Dirichlet Series

242

log(1 + z) =

∞  (−1)n−1

n

n=1

z n for |z| < 1.

This gives −ψ(s) log k =

∞  (−1)n−1

n

n=1

B0−n h(s)n + log B0 + 2iπq.

Let us expand every expression of (h(s))n , and rearrange the terms, which is allowed in the half-plane of absolute convergence of h. We get for ψ an expansion of the form ψ(s) =

∞ 

∞ 

q=0 n 1 ,...,n q

as

  n q −s n 1 −s β(n 1 , . . . , n q ) 1 + c ··· 1 + c . k0 k0 =1

(8.3.6)

This expansion converges absolutely in some half-plane Cθ and can also be written  β(λ)λ−s , ψ(s) = λ∈G(k)

where G(k) denotes the multiplicative semi-group generated by the elements 1 + k cj0 , j ∈ N. To conclude that, in fact, ψ ∈ D even when c0 ≥ 1, we observe that this expansion of ψ holds for every positive integer k and that the intersection all the G(k) s consists of the integers. More precisely: Lemma 8.3.3 The inclusion G(2) ∩ G(3) ⊂ N holds. Indeed, any λ ∈ G(2) ∩ G(3) can be written as λ=

p 2c0 q

=

p 3c0 q 

, 



where p, p  , q, q  are positive integers. Then p3c0 q = p  2c0 q . But 3c0 q is prime with 2c0 q , hence it divides p  by the Gauss lemma, and we are done. All this proves the necessity of the condition in Theorem 8.3.1. We turn to the sufficiency part and start from ϕ(s) = c0 s +

∞ 

cn n −s =: c0 s + ψ(s) ∈ D0 .

(8.3.7)

n=1

The proof is conveniently divided into two steps, even if the first one is a particular case.

8.3 The Arithmetic Theorem

243

1. We have that Cϕ (n −s ) ∈ D for each n ∈ N. Indeed, for n ≥ 1 fixed and for s > σa (ψ), (8.3.7) implies n −ϕ(s) = n −c0 s−c1

∞ 

(−c j log n)k j=2

k!

k≥0

∞   j −ks = n −c0 s−c1 dm m −s ,

(8.3.8)

m=1

 −s a relation which defines the dm = dm(n) ; we set d(s) = ∞ m=1 dm m . By a simple method of majorant series, this implies that |dm | ≤ Dm , where the coefficients Dm are defined through their generating Dirichlet series ∞ 

Dm m −s =

m=1

∞  (|c j | log n)k j=2

k!

k≥0

 j −ks .

(This makes sense, because any integer m ≥ 2 can be written under the form m = jr ≥2, jrkr only in a finite number of ways). kr ≥1 If we fix s > σa (ψ), we therefore obtain ∞ 

Dm m −s =

m=1

∞

∞   exp( j −s |c j | log n) = exp ( j −s |c j |) log n < ∞,

j=2

j=2

 a fortiori ∞ |m −s < ∞, which shows that σa (d) ≤ σa (ψ) < ∞. m=1 |dm −s ∈ H2 , arbitrary. We obtain similarly 2. Let now f (s) = ∞ n=1 an n Cϕ f (s) =

∞ 

an n −ϕ(s) =

n=1

∞ 

an n −c0 s−c1

n=1

∞ 

(−c j log n)k j=2

k!

k≥0

and consequently that (Cϕ f )(s) =

∞ 

j −ks

em m −s ,

m=1

where |em | ≤ E m , the coefficient of m −s in the expansion ∞ 

|an |n

−c0 s−c1

n=1

=

∞ 

∞ 

(|c j | log n)k j=2

k≥0

k!

j −ks



∞ ∞   

|an |n −(c0 s+c1 ) exp log n |c j | j −s = E m m −s .

n=1

Here, we must separate two cases.

j=2

m=1



(8.3.9)

8 Composition Operators on the Space H2 of Dirichlet Series

244

Case 1, c0 ≥ 1. Since ψ ∈ D, we can choose θ real and large enough to ensure that ∞ 

|c j | j −θ =: M < ∞.

j=2

This gives for s = σ + it with σ ≥ θ: ∞ 

E m m −s ≤

∞ 

m=1

|an |n M−(σ+c1 ) < ∞

n=1

as soon as moreover M − σ  − c1 < −σa ( f ) (observe that σa ( f ) < ∞ since −s ∈ D and hence that the Dirichlet series f∈ H2 ⊂ D). This shows that ∞ m=1 E m m ∞ −s e m ∈ D as well. m m=1 Case 2, c0 = 0. This case is slightly more delicate. We then have ϕ = ψ, and we first show a general fact:  −s ∈ Dθ,ν with ψ nonconstant. Then Lemma 8.3.4 Let ψ(s) = ∞ n=1 cn n c1 > ν. Indeed, let N be the first integer ≥2 such that c N = 0. We can write, as s → ∞: 

ψ(s) = c1 + c N N −s + O (N + 1)−s . Let us adjust the real number t0 so as to get c N N −it0 = −|c N |, so that c N N −s = −|c N |N −σ for all s = σ + it0 . This gives us, since we obviously have (N + 1)−σ = oσ (N −σ ) as σ → +∞: 

c1 = ψ(s) + |c N |N −σ + O (N + 1)−σ ≥ ψ(s) > ν for σ large enough, proving the lemma. Now, we claim that in our case c1 >

1 · 2

(8.3.10)

If ϕ is constant, this is obvious. Otherwise, we may use Lemma 8.3.4 with θ = ν = 1/2. We hence get c1 = 1/2 + 2ε with ε > 0. This time, we choose τ real and large enough to ensure (using Lemma 8.3.1 if one wishes) that ∞  j=2

|c j | j −τ ≤ ε.

8.3 The Arithmetic Theorem

245

We get from the preceding, for s real and s ≥ τ : ∞ 

E m m −s ≤

m=1

∞ 

|an |n ε−c1 =

n=1

∞ 

|an |n −1/2−ε < ∞

n=1

 −s where we used Cauchy–Schwarz and the fact that ∞ ∈ H2 . This ends the n=1 an n proof of Theorem 8.3.1.  The proof of Theorem 8.2.1 is much more involved, and will necessitate several additional ingredients. The first one is twisting, a notion which proves to be of basic importance in this analytic theory of Dirichlet series. We accordingly devote a long transition section to that notion. Since we will be dealing with questions of analytic extension from a given open set Ω1 to a given open set Ω2 ⊃ Ω1 (mostly Ω1 = C1/2 and Ω2 = C0 ), let us adopt two terminological conventions for the sequel: • f ∈ H (Ω1 ) can be extended to Ω2 if it has an analytic extension to Ω2 . • ∂Ω1 is a frontier for f ∈ H (Ω1 ) if there is no extension of f to a strictly larger open set Ω2 .

8.4 Twisting 8.4.1 Definition and Uniform Convergence We begin this section with the notion of twisting, which was coined and efficiently used in [11], under the name “vertical limit”. Let M be the set of characters, i.e. the set of all completely multiplicative functions χ : N → T, namely, those satisfying |χ(n)| = 1,

and χ(mn) = χ(m)χ(n) for all m, n ∈ N.

One must have χ(1) = 1. Before proceeding, since twisting happens to be closely related to uniform convergence in half-planes, we prove here a result of independent interest, an improvement of Bohr’s boundedness theorem [9]. Theorem 8.4.1 Let θ, ν be two real numbers, and ψ ∈ Dθ,ν with c0 = 0. Then, the abscissa of uniform convergence of ψ satisfies σu (ψ) ≤ θ. In particular, σc (ψ) ≤ θ. Proof By considering ψ(s + θ) − ν, we may assume θ = ν = 0. Recall the Herglotz theorem ([12], p. 17), according to which every harmonic and non-negative function h on C0 can be written as  h(σ + it) = cσ + Pσ (t − τ )dμ(τ ), (8.4.1) R

246

8 Composition Operators on the Space H2 of Dirichlet Series

c denoting a constant ≥0, μ a positive measure satisfying Poisson kernel of the right half-plane at σ > 0: Pσ (v) =



dμ(τ ) R 1 + τ2

< ∞ and Pσ the

σ 1 · π σ2 + v2

Clearly, 0 < σ ≤ θ ⇒ Pσ (v) ≤

θ Pθ (v) for all v ∈ R. σ

By (8.4.1), a precised Harnack’s inequality ensues 0 < σ ≤ θ ⇒ h(σ + it) ≤

θ h(θ + it). σ

(8.4.2)

Now, since ψ ∈ D, ψ is bounded by a constant M in some half-plane Cθ with θ > 0. Fix 0 < α < 1 and apply (8.4.2) to the non-negative harmonic function h = ψ α . This function moreover satisfies (on C0 ) |ψ|α ≤

1 h =: Cα h. cos(απ/2)

We thus obtain, for 0 < σ ≤ θ, the inequalities |ψ(σ + it)|α ≤ Cα h(σ + it) ≤ Cα

θ θ θ h(θ + it) ≤ Cα |ψ(θ + it)|α ≤ Cα M α . σ σ σ

Hence, ψ is bounded in Cε for all ε > 0. By Bohr’s boundedness Theorem 6.2.3, we get σu (ψ) ≤ ε, and finally σu (ψ) ≤ 0 since ε > 0 is arbitrary.  −s Now, let f : Cθ → Cν analytic, and satisfying f (s) = ∞ ∈ D. We can n=1 an n formally associate to f , for every character χ ∈ M, the twisted function f χ (s) =

∞ 

an χ(n)n −s .

(8.4.3)

n=1

We extend the definition to all functions f (s) = c0 s + F(s) ∈ Dθ,ν by setting f χ (s) = c0 s + Fχ (s).

(8.4.4)

A typical example is that of χ(n) = n −iτ where τ ∈ R, and in that case Fχ (s) = F(s + iτ ) =: Fτ (s), a so-called vertical translate of F, which we already encountered in Chap. 7. We set, according to (8.4.4): f τ (s) = c0 s + Fτ (s),

8.4 Twisting

247

and also call it, with an abuse of language, a vertical translate of f , even though f τ (s) is not equal to f (s + iτ ) when c0 ≥ 1. Those vertical translates generate in some sense all twisted functions, as indicated by the simple Proposition 8.4.1 Fix f (s) = c0 s + F(s) ∈ Dθ,ν . Then, in the half-plane Cθ , the twisted functions f χ are exactly the limits (in the sense of uniform convergence on compact subsets of Cθ ) of some sequence ( f τ N ) of vertical translates of f , and σu (Fχ ) ≤ θ. Proof We can ignore the common linear part c0 s. Suppose first that some sequence (Fτ N ) converges to g ∈ H (Cθ ). Taking a subsequence of (τ N ) if needed, we can ∈ N and clearly χ ∈ M. Fix θ > θ and let assume that n −iτ N → χ(n) for each n −s on Cθ , we can find an integer P(ε) ε > 0. By the uniform convergence of ∞ 1 an n such that q     an n −iτ N n −s  ≤ ε for all s ∈ Cθ and all q > p > P(ε).  p+1

Passing to the limit as N → ∞, we obtain q     an χ(n)n −s  ≤ ε for all s ∈ Cθ and all q > p > P(ε).  p+1

It follows that σu (Fχ ) ≤ θ and that Fτ N converges to Fχ in Cθ . Hence, g = Fχ . Let conversely χ ∈ M. By Theorem 2.2.6 in Chap. 2, a corollary to Kronecker’s theorem, for each integer N we can find a real number τ N so that |χ(n) − n −iτ N | ≤ 1/N , n = 1, . . . , N . In particular, lim N →∞ n −iτ N = χ(n) for each n ∈ N. As above, this implies that σu (Fχ ) ≤ θ and that Fτ N converges to Fχ in Cθ . Here is a first short list of properties of twisting: Proposition 8.4.2 Fix χ ∈ M, F, G ∈ D, and ϕ(s) = c0 s + ψ(s) ∈ Dθ,ν . Then 1. If σa (F) ≤ θ and σu (G) ≤ θ, (F G)χ = Fχ G χ in Cθ (product rule). 2. (n −c0 s )χ = χ(n)c0 n −c0 s . 3. (n −ϕ )χ = χ(n)c0 n −ψχ in the half-plane Cθ . Proof For the first item, use the Dirichlet product rule, the inequality σu (F G) ≤ max(σa (F), σu (G)) (see Exercise 2) and the complete multiplicativity of χ. For the second one, just use n −c0 s = (n c0 )−s . For the third one, use n −ϕ(s) = n −c0 s n −ψ(s) , the fact that (n −ψ )χ = n −ψχ

248

8 Composition Operators on the Space H2 of Dirichlet Series

in the half-plane of uniform convergence of the Dirichlet series ψ (this is obvious for vertical translates, then use Proposition 8.4.1), and the product rule above.

8.4.2 Mapping Properties Here is next a longer list of the mapping properties of the Dirichlet part ψ of a symbol ϕ(s) = c0 s + ψ(s) ∈ Dθ,ν . The twisted part ψχ is also considered. Those properties will reveal very useful in the proof of our main theorem. Proposition 8.4.3 Assume that the analytic function ϕ satisfies ϕ(s) = c0 s + ψ(s) ∈ Dθ,ν . Then: 1. If ψ is a constant c1 , this constant belongs to Cν−c0 θ . 2. If ψ is nonconstant, it maps Cθ to Cν−c0 θ and moreover 3. for all θ > θ, ψ maps Cθ to Cν+ε−c0 θ with ε = ε(θ ) > 0 and ψ is bounded from above on Cθ . 4. One can everywhere replace ψ by ψχ where χ ∈ M. Proof 1, 2. We first claim that 2−ψ is bounded in Cθ .

(8.4.5)

Indeed, fix θ > θ and take an element s ∈ ∂Cθ , i.e. s = θ . Then ψ(s) = ϕ(s) − c0 s > ν − c0 θ . Besides, ψ is uniformly bounded in Cθ since its Dirichlet series converges uniformly in that half-plane by Theorem 8.4.1. So that, using the maximum modulus principle (for bounded analytic functions in arbitrary domains):  |2−ψ | ≤ 2c0 θ −ν throughout Cθ . Now, letting θ tend to θ, we obtain |2−ψ | ≤ 2c0 θ−ν throughout Cθ , that is, ψ(s) ≥ ν − c0 θ throughout Cθ . If ψ is nonconstant, the open mapping theorem gives the strict inequality. 3. Next, we show that if ψ is nonconstant, ψ maps Cθ to some smaller half-plane Cν+ε−c0 θ with ε = ε(θ ) > 0. For that purpose, we set F(s) = 2−ψ(s) and, for x ≥ θ: M(x) = M F (x) = sup{|F(s)| : s > x} < ∞.

8.4 Twisting

249

As we have shown, M(θ) ≤ 2c0 θ−ν =: m. Let us prove the strict inequality M(A) < m for A large enough. Indeed, in view of 1., we may apply Lemma 8.3.4, which gives us c1 ≥ ν + ε − c0 θ for some ε > 0. And we then see that inf [ψ(A + it)] ≥ ν +

t∈R

ε − c0 θ 2

for A large enough, implying M(A) < m. The idea is now to use a convexity property of M to show that indeed M(θ ) < m as soon as θ > θ. To that effect, we recall that, by the three lines theorem, M is logarithmically convex on [θ, ∞[. Hence, if θ = (1 − λ)θ + λA with 0 < λ < 1, we get M(θ ) ≤ M(θ)1−λ M(A)λ < m 1−λ m λ = m. This gives the conclusion. 3. It remains to see that the function ψ is bounded from above in Cθ when θ > θ. Clearly, 2−ψ ∈ D since ψ ∈ D, hence some horizontal translate of it lies in H2 and we can apply Theorem 8.3.1. We saw that 2−ψ is moreover bounded on Cθ , equivalently that ψ is bounded from below on Cθ . By the Bohr boundedness theorem, the Dirichlet series corresponding to 2−ψ converges uniformly on each half-plane Cθ with θ > θ, or else σu (2−ψ ) ≤ θ. Suppose that ψ is not bounded in Cθ , then ψ(sn ) → +∞, equivalently 2−ψ(sn ) → 0, along some sequence sn = σn + iτn with σn ≥ θ . Since ψ(s) → c1 as s → ∞, we must have σn ≤ C for some constant C. But since σu (ψ) ≤ θ < θ , we can apply Proposition 8.4.1 to say that, taking a subsequence if necessary, the vertical translates ψτn converge to some twisted function ψχ and that σn → σ ∈ [θ , C]. Passing to the limit in (2−ψ )τn (σn ) = 2−ψ(sn ) → 0, we get that (2−ψ )χ (σ) = 0, or again, using Proposition 8.4.2, that 2−ψχ (σ) = 0, which is obviously impossible. 4. That we may replace ψ by ψχ in the statement follows from Proposition 8.4.2 applied to ψ instead of ϕ, once we have shown that this is licit, equivalently that ψχ is bounded from above on Cθ . And this is easy, since |2−ψ | ≥ δ > 0 on Cθ (using 3.), we again have, for its vertical limit function, |2−ψχ | ≥ δ > 0 on Cθ , and the conclusion follows. A useful corollary (see [13] for a more precise version) is Proposition 8.4.4 Let ϕ ∈ G. If ϕ(s) = s + iτ with τ a real constant, then there exists ε > 0 such that ϕ(C1/2 ) ⊂ C1/2+ε . Proof We separate three cases: 1. If c0 = 0, we know that ϕ(C0 ) ⊂ C1/2 . If ψ is constant, the result is obvious; if not, Proposition 8.4.3 with θ = 0, θ = 1/2, ν = 1/2 gives the result.

8 Composition Operators on the Space H2 of Dirichlet Series

250

2. If c0 = 1 with ψ = c1 , a constant, the result is clear since c1 ≥ 0 and c1 = 0 by hypothesis. Otherwise, ψ(C1/2 ) ⊂ Cε for some ε > 0 by Proposition 8.4.3 with θ = ν = 0, θ = 1/2, so that ϕ(C1/2 ) ⊂ C1/2+ε . 3. If c0 > 1, since ψ(C0 ) ⊂ C0 , we get that ϕ(C1/2 ) ⊂ C c20 ⊂ C1 .

8.4.3 Twisting and Composition Since twisting is intended to help us describing the composition operators Cϕ : H2 → H2 , the following proposition is important. Proposition 8.4.5 Let θ ≥ 0 and be a nonconstant,  let ϕ(s)−s= c0 s + ψ(s) ∈ Dθ,1/2 2 c n . Then, for f ∈ H and χ ∈ M, the folanalytic function, with ψ(s) = ∞ n n=1 lowing relation holds:

 f ◦ ϕ χ (s) = f χc0 ◦ ϕχ (s) for all s ∈ Cθ .

(8.4.6)

Proof We note that the relation (8.4.6) makes sense. Indeed, we know from Proposition 8.4.3 that the twisted function ϕχ also maps Cθ to C1/2 . Since f χc0 ∈ H2 when the RHS of (8.4.6) defines a holomorphic function on Cθ . Moreover, if f ∈ H2 , −s f (s) = ∞ n=1 an n , with absolute convergence on C1/2 , one can write

∞   f ◦ ϕ (s) = an n −ϕ(s) , s ∈ Cθ .

(8.4.7)

n=1

We first claim (a variant of Proposition 8.4.4) that s > θ > θ =⇒ ϕ(s) ≥ 1/2 + ε for some ε = ε(θ ) > 0.

(8.4.8)

Indeed, if c0 = 0, ϕ = ψ is nonconstant, and we use 3. of Proposition 8.4.3. If c0 ≥ 1, let s ∈ Cθ . We get using 2. of Proposition 8.4.3, ϕ(s) = c0 s − ψ(s) ≥ c0 (s − θ) + 1/2 ≥ c0 (θ − θ) + 1/2. Denote now by . H ∞ (Cθ ) the sup-norm in Cθ . Since f ∈ H2 , we infer from (8.4.8) and Cauchy–Schwarz that ∞ 

|an | n −ϕ H ∞ (Cθ ) < ∞.

(8.4.9)

n=1

Let f N (s) =

N n=1

an n −s be a partial sum of f . Proposition 8.4.2 implies

8.4 Twisting

251

( f N ◦ ϕ)χ (s) =

N 

an χ(n)c0 n −ϕχ (s) = ( f N )χc0 ◦ ϕχ (s), s ∈ Cθ .

n=1

We pass to the limit as N → ∞, using that f N ◦ ϕ converges uniformly to f ◦ χ on Cθ by (8.4.9), and that the operation of twisting is continuous with respect to the uniform norm in Cθ . This gives ( f ◦ ϕ)χ (s) =

∞ 

an χ(n)c0 n −ϕχ (s) = ( f )χc0 ◦ ϕχ (s), s ∈ Cθ .

n=1

Since θ > θ is arbitrary, the result follows.

8.4.4 Twisting and Probability A random multiplicative character will be a character such that the χ( p), for p prime, are independent and uniformly distributed on the circle T (Steinhaus variables). The underlying probability space corresponds to (Ω, A, P) with Ω = p T p where p runs once and for all over primes, T p is a copy of T and P = ⊗ p m where m is the Haar measure of T. Then, the values χ(n), with χ(n) =



α χ( p) , pα ||n

are no longer independent, but are clearly orthonormal (see Exercise 3). We accordingly recall a classical almost sure convergence theorem. Theorem 8.4.2 (Menchoff) Let (X n ) be an orthonormal sequence in a probability space (Ω, A, P) and let (an ) be a sequence of complex numbers such that ∞ 

|an |2 log2 n < ∞.

n=1

Then, the series

∞ n=1

an (X n ) converges almost surely.

This theorem is due to Menchoff (see [14] or [15] for detailed proofs, which are ∞tricky 2 but elementary). If the X n are moreover independent, the condition n=1 |an | < ∞ suffices to ensure the almost sure convergence of the implied series (Khintchine–Kolmogorov theorem). If we only assume orthogonality, we must add this extra factor log2 n, which curiously enough, as proved by Menchoff and Tandori, turns out to be optimal in full generality ([14, 16] Chap. VIII) (see also the remark below). A useful property which we mention in passing is

8 Composition Operators on the Space H2 of Dirichlet Series

252

If c0 ∈ N, the map χ → χc0 : M → M preserves the measure P.

(8.4.10)

This is obvious, since z → z c0 : T → T preserves the Haar measure m. A second connection between extending and twisting, also in terms of frontier, is the following (see [11]).  −s ∈ H2 . Then, almost surely, σc ( f χ ) ≤ 0 and Theorem 8.4.3 Let f (s) = ∞ n=1 an n hence  the twisted function f χ can be extended to C0 . Moreover, if f (s) = p a p p −s ∈ H2 , and if σc ( f χ ) = 0, the line s = 0 is almost surely a frontier for f χ .  2 −2ε log2 n < ∞. By Menchoff’s Theorem Proof Fix ε > 0. have ∞ n=1 |an | n We ∞ −ε 8.4.2, the series n=1 an χ(n)n converges almost surely and σc ( f χ ) ≤ ε. The first part clearly follows. For the second part, we note that the variables χ( p) are independent and symmetric, and we can use a result of Kahane ([17], Theorem 4.4, p. 44). Remark In [18], a stronger result is proved, namely, 

|an |2 < ∞ =⇒



an χ(n) converges almost surely.

No extra factor log2 n is needed here. But the proof is much more involved and uses Carleson’s almost everywhere convergence theorem, which it contains! Indeed, the result can be seen as a bridge between the Khintchine–Kolmogorov theorem (take an = 0 if n is not a prime) and Carleson’s theorem (take an = 0 if n is not a power of 2). And this involved result is not needed for proving Theorem 8.4.3.

8.4.5 Twisting and Topology A variant of random

twisting will be needed in the proof. Essentially, we consider the space Ω = p T of the previous subsection as a compact metric space (and hence a Baire space, i.e. every countable union of closed sets with empty interior has empty interior again) and no longer as a probability space, and this makes a serious difference. For example, if the a p are scalars, and with the terminology below, we have (see Exercise 4)   a p χ( p) converges almost surely ⇐⇒ |a p |2 < ∞. p



a p χ( p) converges quasi-surely ⇐⇒



|a p | < ∞.

p

The variant considered here is the following (we just change notations, see also ∗ [10]): let Ω = TN (a compact, metrizable, set if equipped with the topology of

8.4 Twisting

253

simple convergence), whose elements are denoted ω = (Zn (ω)). It than is∞ simpler −s a n into the twisting model in the probabilistic sense, because we twist n=1 n ∞ −s a Z n where the Z are P-independent complex choices of signs (but in n n=1 n n replacing χ(n) by Z n , we drop the multiplicativity property). But here we use as said above a topological framework (Baire’s theorem). will say that E ⊂ Ω is quasi We −λ ns b e , a general Dirichlet series sure if E contains a dense G δ set. Let f (s) = ∞ n=1 n (i.e. λn ≥ 0 and λn increases to +∞) with absolute convergence abscissa σa = α. We set, for ω ∈ Ω, f ω (s) =

∞ 

Z n (ω)bn e−λn s , with again σa ( f ω ) = α.

n=1

With this terminology, the following holds (see, for example, [17], or also [19], with a misprint, p. 270).  −λn s Theorem 8.4.4 Let f (s) = ∞ be a general Dirichlet series with abson=1 bn e Then, the line s = α is quasi-surely a fronlute convergence abscissa σa = α ∈ R. tier for f ω . In particular, for f (s) = p a p p −s supported by the primes, the line s = α is quasi-surely a frontier for f χ . Proof Let Q denote the set of rational numbers, and E the set of those ω ∈ Ω such that s = α is not a frontier for f ω . Clearly,

E=

E a,r,N , where a = α + it, t ∈ Q, r ∈ Q+ , N ∈ N

a,r,N

 −λn s and where E a,r,N is the set of ω’s such that ∞ can be analytically n=1 bn Z n (ω)e continued as f ω in D(a, r ) = {s ; |s − a| < r } with | f ω | ≤ N in D(a, r ). A normal family argument shows that E a,r,N is closed in Ω. If some point ω0 = (z n ) is interior to E a,r,N , let M be an integer such that, if Z n (ω  ) = z n for n ≤ M, then ω  ∈ E a,r,N . For any ω ∈ Ω, one can write f ω (s) =

M  n=1

z n bn e

−λn s

+

∞ 

Z n (ω)bn e

−λn s



n=M+1

M   + (Z n (ω) − z n )bn e−λn s n=1

=: f ω (s) + P(s) One can hence extend analytically with ω  ∈ E a,r,N and P a Dirichlet polynomial. M |bn |e−λn (α−r ) =: C. Fix 0 < ρ < r/3 so f ω to D(a, r ) with | f ω (s)| ≤ N + 2 n=1 that D(a + ρ, 2ρ) ⊂ D(a, r ). The Cauchy inequalities and a permitted term-by-term differentiation (since a + ρ ∈ Cα ) give ∞  C 1   | f ω( j) (a + ρ)|  = bn Z n (ω)λnj e−λn (a+ρ)  ≤ , j = 0, 1, . . .  j! j! n=1 (2ρ) j

8 Composition Operators on the Space H2 of Dirichlet Series

254

Since this holds for every choice of complex signs ω, ∞

1  C |bn |λnj e−λn (α+ρ) ≤ · j! n=1 (2ρ) j

(8.4.11)

If we now fix ρ < R < 2ρ, multiply each term of (8.4.11) par R j , sum up and permute, we get ∞ 

|bn |e−λn (α+ρ−R) ≤ C

n=1

∞   R j < ∞. 2ρ) j=0

But this is impossible since α + ρ − R < α. This contradiction shows that all closed sets E a,r,N have empty interior, as well as their countable union, by Baire’s theorem. The second part follows, because the χ( p) are nothing but independent choices of complex signs. All this ends the proof. We will need the following corollary, where p once more denotes a prime.  Corollary 8.4.1 There is a Dirichlet series f (s) = p a p p −s , with square2 a character χ0 ∈ M such that the twisted summable coefficients a p (i.e.  f ∈ H ) and Dirichlet series f χ0 (s) = p a p χ0 ( p) p −s satisfies σa ( f χ0 ) = 1/2 and has the vertical line s = 1/2 as a frontier.  Proof One applies the second part of Theorem 8.4.4 to f (s) = p a p p −s , with  a p = √ p 1log p , which satisfies σa ( f ) = 1/2. If f χ (s) = p χ( p)a p p −s , one still has σa ( f χ ) = 1/2, and one can find χ0 such that the line s = 1/2 is a frontier for f χ0 . Now, the series f χ0 satisfies both conditions required since  p

|a p |2 =

 p

1 < ∞, p log p

and σa ( f χ0 ) = 1/2.

8.5 Integral Representation and Embedding 8.5.1 Integral Representation It is useful to dispose of an asymptotic integral formula for the H2 -norm, analogous to Parseval’s formula for the Hardy space H 2 of the disc. Several such formulas exist, especially of Littlewood–Paley type, valid also for H p -spaces of Dirichlet series [13, 20, 21]. The following one will suffice to our purposes here.  Theorem 8.5.1 Let f (s) = n≥1 an n −s ∈ H2 , where the Dirichlet series converges uniformly on C0 . Let u ∈ L 1 (R) such that u ≥ 0 and u 1 = 1, with dilates u a (t) = a1 u( at ), a > 0. Then

8.5 Integral Representation and Embedding

255



f 2H2 = lim

a→∞ R

| f (it)|2 u a (t)dt.

Proof If f is a Dirichlet polynomial, i.e. am = 0 for large m, we can give a formal u (aξ) and  u (0) = 1 to get: proof. Namely, we use ua (ξ) =   R

| f (it)|2 u a (t)dt =



 am an

m,n

=



R



−it

 m/n u a (t)dt = am an ua log(m/n) m,n

∞   a→∞  am an  u a(log m − log n) −→ |an |2 = f 2H2

m,n

n=1

by the Riemann–Lebesgue lemma. In the general case, let ε > 0 and N0 a positive integer satisfying supt∈R | f (it) − f N (it)| ≤ ε for N ≥ N0 , where f N (s) =  N −s n=1 an n . The triangle inequality, and R u a (t)dt = 1, gives for N ≥ N0 (writing μa = u a (t)dt and f, f N for f (it), f N (it)):  R

| f |2 dμa

1/2



≤

R

| f − f N |2 dμa

≤ε+

 R

1/2

| f N |2 dμa

+

 R

| f N |2 dμa

1/2

1/2

whence by the preceding  lim sup a→∞

R

| f |2 dμa

1/2

≤ε+

N



|an |2

1/2

≤ ε + f H2 .

n=1

Similarly, for N ≥ N0 ,  lim inf a→∞

R

| f |2 dμa

1/2

≥ −ε +

N



|an |2

1/2

n=1

and the result ensues by letting N tend to ∞ and ε to 0.

8.5.2 Embedding Results for H2 Let us now describe an ingredient (a consequence of the Montgomery–Vaughan inequality of Chap. 1) to intervene in the proof of the sufficiency in Theorem 8.2.1, which helps to make the connection between both spaces H 2 (C1/2 ) and H2 [22]. This ingredient seems essential, since its failure for the spaces H p , 0 < p < 2,

8 Composition Operators on the Space H2 of Dirichlet Series

256

recently proved by A. Harper (we will come back to this) will also imply the failure of Theorem 8.2.1 for those spaces. ∞ −s Theorem ∞ 8.5.22 (Local embedding) Let f (s) = n=1 an n be a Dirichlet series with n=1 |an | < ∞. One has uniformly with respect to k ∈ R and ε > 0:  k

k+1

∞     f (1/2 + ε + it)2 dt  |an |2 = f 2H2 .

(8.5.1)

n=1

Proof Just apply the Montgomery–Vaughan inequality of Chap. 1, with an changed into an n −1/2 , in the special case λn = log n and hence δn ≈ 1/n. A useful variant of the local embedding is (cf. [22]).4  −s Theorem 8.5.3 (Global embedding) Let f (s) = ∞ ∈ H2 . n=1 an n 2 F(s) = f (s)/s belongs to the usual Hardy space H (C1/2 ) and moreover

F 2H 2 (C1/2 ) ≤ C f 2H2 ,

Then,

(8.5.2)

where C is a numerical constant. Equivalently, for each b > 0, there exists a positive constant Cb such that, if f ∈ H2 :  R



  f 1/2 + it 2 Pb (t)dt ≤ Cb f 2 2 , H

where Pb denotes the Poisson kernel of C0 at b, Pb (t) =

(8.5.3)

b · π(b2 +t 2 )

Proof Note that (8.5.2) implies, via Fatou’s theorem for H 2 (C1/2 ), the almost everywhere existence of the radial limit



 f 1/2 + it = lim f 1/2 + ε + it ε→0

as f ∈ H2 . Let us now establish formula (8.5.3). We first use (8.5.1) which gives us, for ε > 0:  k+1    f (1/2 + ε + it)2 dt ≤ C f 2 2 H

k

where C is a numerical constant. Next  | f (1/2 + ε + it)| Pb (t)dt = 2

R

≤

 k∈Z

k∈Z k+1

k



k+1

| f (1/2 + ε + it)|2 Pb (t)dt

k

 Cb | f (1/2 + ε + it)|2 dt ≤ 2 k +1 k∈Z

 k

k+1

Cb

f 2H2 ≤ Cb f 2H2 , k2 + 1

8.5 Integral Representation and Embedding

257

where the implied constants Cb only depend on b. We finish by letting ε tend to 0, and applying Fatou’s lemma.

8.6 Proof of the Main Theorem We are now ready to prove the following precised form of the main theorem, with the slight improvement concerning σu (ψ): Theorem 8.6.1 Let ϕ : C1/2 → C1/2 be analytic. The following are equivalent: 1. Cϕ : H2 → H2 . 2. ϕ ∈ G, ϕ(s) = c0 s + ψ(s). Moreover • The operator Cϕ is a contraction iff c0 ≥ 1. • In this case, σu (ψ) ≤ 0 (and σu (ψ) ≤ 1/2 when c0 = 0). The proof will be once again conveniently split into several parts.

8.6.1 Proof of Necessity in Theorem 8.6.1 We can already observe that ϕ ∈ D0 in view of the algebraic Theorem 8.3.1, but we will need more. It is convenient to treat separately the cases c0 ≥ 1 and c0 = 0.

8.6.1.1

The Case c0 ≥ 1

One of the difficulties, compared to the usual Hardy case, is that the identity map does not belong to our space H2 , so we do not have the pleasant starting point ϕ ∈ H 2 because ϕ = Cϕ (z) or because ϕ ∈ H ∞ . First, we establish a lemma which is a prototype of the additional efforts needed here. Lemma 8.6.1 Let ϕ be analytic function on C1/2 . Assume that n −ϕ has an extension f n to an open simply connected superset Ω of C1/2 for n = 2, 3. Then, ϕ itself can be extended to Ω. Proof Since

f n (s) = −ϕ (s) log n, s ∈ C1/2 f n (s)

(8.6.1)

the left-hand side provides a meromorphic extension of ϕ to Ω. Fix s0 ∈ Ω, and let kn ≥ 0 be the order at which the function f n vanishes at s0 . Denoting by ρ(s0 ) the residue of ϕ at s0 and taking residues in (8.6.1), we get (n = 2, 3)

8 Composition Operators on the Space H2 of Dirichlet Series

258

kn = − log n × ρ(s0 ). But then ρ(s0 ) = 0 otherwise log 2/ log 3 would be rational. Hence, ϕ can be extended to Ω, and ϕ as well, since Ω is simply connected. Now, assume that Cϕ : H2 → H2 is bounded. We know, as said before, that ϕ ∈ D0 , but more precisely Lemma 8.6.1 applied to Ω = C0 , and to the functions f n = n −ϕχ = (n −ϕ )χ tells that ϕχ can be almost surely extended to C0 . This being said, fix f ∈ H2 . Theorem 8.4.3, (8.4.10) and the preceding show that, for almost χ (say χ ∈ E), the three functions ( f ◦ ϕ)χ , f χc0 , ϕχ can be extended to C0 , with moreover (using Proposition 8.4.5), for every χ ∈ E: ( f ◦ ϕ)χ (s) = f χc0 ◦ ϕχ (s), s ∈ C1/2 .

(8.6.2)

We next wish to show that ϕχ (C0 ) ⊂ C0 for some χ ∈ E, i.e. Ω = C0 where Ω = {s ∈ C0 : ϕχ (s) > 0}, an open set containing C1/2 by Proposition 8.4.3. Let Ω0 be the connected component of C1/2 in Ω. If Ω = C0 , Ω0 = C0 as well; by connectedness, Ω0 is not closed in C0 and there exists s0 ∈ ∂Ω0 ∩ C0 , that is, s0 > 0 and ϕχ (s0 ) = 0. We claim that we can moreover assume ϕχ (s0 ) = 0. For that purpose, let Z be the discrete zero set of ϕχ in C0 , and B ⊂ C0 a small ball of radius r centred at s0 . Since h := ϕχ is harmonic in C0 , it holds  h(x + i y)d xd y = πr 2 h(s0 ) = 0, B

and we can find u, v ∈ B \ Z such that h(u) < 0 and h(v) > 0. Let γ ⊂ B be a curve joining u to v and disjoint from Z . By the intermediate value theorem, there is w ∈ γ such that h(w) = 0 and we can replace s0 by w if needed. Now, since ϕχ (s0 ) = 0, the map ϕχ is conformal near s0 and, by analytic continuation, (8.6.2) holds on Ω0 , so that the formula f χc0 = ( f ◦ ϕ)χ ◦ ϕ−1 χ provides an analytic extension of f χc0 near the point s1 = ϕχ (s0 ) ∈ ∂C0 . Finally, suppose that we can find f ∈ H2 such that ∂C0 is almost surely (say χ ∈ F) a frontier for f χc0 . If we start from this f in our reasoning, we will get a contradiction for some χ ∈ E ∩ F. We hence proved that ϕχ (C0 ) ⊂ C0 for some χ ∈ M. This implies ψχ (C0 ) ⊂ C0 by Proposition 8.4.3.

8.6 Proof of the Main Theorem

259

We may then apply Theorem 8.4.1 to get σu (ψχ ) ≤ 0. In turn, we may apply Proposition 8.4.3 again, and untwist ψχ by writing ψ = (ψχ )χ , to get ψ(C0 ) ⊂ C0 , . We are done, because Theorem 8.4.3 provides the desired f hence ϕ(C0 ) ⊂ C0 (take, e.g. f (s) = p p −1−s )! 8.6.1.2

The Case c0 = 0

In this case, ϕ = ψ, and since ϕ(C1/2 ) ⊂ C1/2 , Theorem 8.4.4 applies and shows that σu (ϕ) ≤ 1/2. And (8.6.2) then reads, for all f ∈ H2 and some χ = χ f ∈ M: ϕχ (C1/2 ) ⊂ C1/2

and ( f ◦ ϕ)χ (s) = f ◦ ϕχ (s), s ∈ C1/2 .

(8.6.3)

If we can find f ∈ H2 such that ∂C1/2 is a frontier for f , the same proof as in the case c0 ≥ 1, starting this time from Ω = {s ∈ C0 : ϕχ (s) ∈ C1/2 }, will show that such that ϕχ (C0 ) ⊂ C1/2 , and (8.6.3) will hold for all s ∈ C0 by analyticity. We will again be able to untwist this ϕχ to conclude that ϕ has an analytic extension to C0 satisfying ϕ(C0 ) ⊂ C1/2 . But the existence of such an f is given this time by the topological twisting, namely, by Corollary 8.4.1 of Theorem 8.4.4. We are again done.

8.6.2 Proof of Sufficiency in Theorem 8.6.1 8.6.2.1

The Case c0 ≥ 1

Proof We first consider a simpler case. Let f be a Dirichlet polynomial, f (s) =

N 

bn n −s .

n=1

We now proceed in three steps. 1. We have that Cϕ f ∈ D, the ring of convergent Dirichlet series: since by hypothesis ϕ ∈ G ⊂ D0 , this follows from Theorem 8.3.1. 2. Moreover, Cϕ f ∈ H ∞ (C0 ). Indeed, using that ϕ(C0 ) ⊂ C0 , we get | f [ϕ(s)]| ≤ N n=1 |bn | for s ∈ C0 . Therefore, Cϕ f ∈ H∞ , a fortiori Cϕ f ∈ H2 (those two steps will remain valid when c0 = 0). 3. We are left with the task of controlling the norm Cϕ f H2 more accurately. To this effect, we consider the fractional linear map Ta : C0 → D defined for a > 0 by

8 Composition Operators on the Space H2 of Dirichlet Series

260

Ta (s) =

a−s a+s

with Ta−1 (z) = a

1−z : D → C0 , Ta−1 (eiu ) = −ia tan(u/2). 1+z

The change of variable a tan(u/2) = t, u = 2 arctan(t/a) shows that, for every test function h:    π  −1 iu  1 h Ta (e ) du = h(−it)Pa (t)dt = h(it)Pa (t)dt, (8.6.4) 2π −π R R where Pa is the (already used) Poisson kernel at a of the right half-plane C0 , Pa (t) =

a 1 t = u( ) π(a 2 + t 2 ) a a

with u(t) =

1 · π(1 + t 2 )

For a, b, ε > 0, let ϕε (s) = ϕ(s + ε), well defined since ϕ has an analytic extension to C0 , and let ω = Tb ◦ ϕε ◦ Ta−1 : D → D, as well as F = f ◦ Tb−1 . We have N F ∈ H ∞ (D) because |F(z)| ≤ n=1 |bn |, a fortiori F ∈ H 2 and (8.6.4) gives us 1 2π



π

−π

 |F(eiu )|2 du =

R

| f (it))|2 Pb (t)dt.

(8.6.5)

Now, Littlewood’s subordination principle ([1], p. 16, see also Theorem 8.1.1) tells that  π  π 1 1 + |ω(0)| 1 |F ◦ ω(eiu )|2 du ≤ |F(eiu )|2 du. (8.6.6) 2π −π 1 − |ω(0)| 2π −π Or again, using (8.6.4) and (8.6.5), since F ◦ ω = f ◦ ϕε ◦ Ta−1 :  R

| f ◦ ϕε (it)|2 Pa (t)dt ≤

1 + |ω(0)| 1 − |ω(0)|

 R

| f (it)|2 Pb (t)dt.

(8.6.7)

We next observe that f and f ◦ ϕε have uniformly convergent Dirichlet series in C0 . This is obvious for f ; besides, ( f ◦ ϕε )(s) =

∞  n=1

βn n −s−ε

if ( f ◦ ϕ)(s) =

∞ 

βn n −s

n=1

and Bohr’s boundedness theorem (applicable f ◦ ϕ belongs to H∞ ) guarantees ∞ since −s−ε in C0 , ε > 0. Let us now pass the uniform convergence of the series n=1 βn n to the limit in (8.6.7) as a → ∞, adjusting b = c0 a, and using Theorem 8.5.1 with −1

u(t) = π(1 + t 2 ) . This is licit since f and Cϕε f have uniformly convergent Dirichlet series in C0 . Moreover,

8.6 Proof of the Main Theorem

ω(0) =

261

1 − b/ϕε (a) a→∞ −→ 0 1 + b/ϕε (a)

since ϕε (a) ∼ c0 a = b.

All this gives f ◦ ϕε 2H2 ≤ f 2H2 and then

f ◦ ϕ 2H2 ≤ f 2H2 by letting ε tend to zero. This inequality is still valid, by approximation,   N for all−sfunc−s 2 b n ∈ H and if f (s) = tions in H2 . In fact, if f (s) = ∞ n N n=1 n=1 bn n , the w 2 2 2 preceding implies that Cϕ f N H ≤ f N H ≤ f H whence Cϕ f N −→ g ∈ H2 uc with necessarily g = Cϕ f since one has f N ◦ ϕ −→ f ◦ ϕ. So that Cϕ f ∈ H2 , with

Cϕ f H2 ≤ lim inf Cϕ f N H2 ≤ f H2 . N →∞

In particular, Cϕ is a contraction.

8.6.2.2

The Case c0 = 0

Suppose finally that c0 = 0. Let S : C1/2 → C0 be the shift defined by S(s) = s − 21 · The diagram (justified by the extra assumption ϕ(C0 ) ⊂ C1/2 ) Ta−1

ϕ

Tb

S

D −→ C0 −→ C1/2 −→ C0 −→ D indicates that ω = Tb ◦ S ◦ ϕ ◦ Ta−1 maps D to itself and that ω(0) = Tb [ϕ(a) − 1/2]. If f is a Dirichlet polynomial and F = f ◦ S −1 ◦ Tb−1 , it holds F ◦ ω = f ◦ ϕ ◦ Ta−1 , and designating by m the normalized Haar measure of T, Littlewood’s subordination |ω(0)| = Ma,b : principle once more gives, setting 11 + − |ω(0)|  T

| f ◦ ϕ ◦ Ta−1 |2 2 dm ≤ Ma,b

 T

| f ◦ S −1 ◦ Tb−1 |2 2 dm

and consequently, using (8.6.4), and Theorem 8.5.3 in an essential way: 

 |( f ◦ ϕ)(it)| Pa (t)dt ≤ Ma,b 2

R

R

| f (1/2 + it)|2 Pb (t)dt ≤ Cb Ma,b f 2H2 ·

262

8 Composition Operators on the Space H2 of Dirichlet Series

Let us pass to the limit in this inequality, as a → ∞, keeping this time b fixed. Since the Dirichlet series of f ◦ ϕ is uniformly convergent in C1/2 and since a→∞ Ma,b −→ Mb := (1 + |Tb (c1 − 1/2)|)(1 − |Tb (c1 − 1/2)|)−1 (recall that c0 = 0 and that c1 > 1/2), we get as in the case c0 ≥ 1 (there again, one should first consider ϕε and then let ε tend to 0, we skip that step):

f ◦ ϕ 2H2 ≤ Cb Mb f 2H2 . By approximation, this inequality extends to an arbitrary function f ∈ H2 . Let us finally show that Cϕ is not a contraction, and that more precisely:  1/2

Cϕ ≥ ζ(2 c1 ) > 1.

(8.6.8)

Indeed, K b (w) = ζ(w + b) designating the reproducing kernel at b ∈ C1/2 of H2 , the mapping equation (8.1.2) gives for a > 1/2: ζ(2 ψ(a)) = K ϕ(a) 2H2 = Cϕ∗ (K a ) 2H2 ≤ Cϕ 2 K a 2H2 = Cϕ 2 ζ(2a). Letting a tend to +∞, one gets the result, since ψ(a) → c1 , ζ(2a) → 1 and since ζ maps ]1, ∞[ to itself. The proof is finished. Remark Other Hilbert spaces of Dirichlet series, the spaces of Bohr–Bergman type, as well as their composition operators, have been recently studied by Bailleul and Brevig [23]. See also [24]. We highly recommend those articles to the reader. Here are now some simple examples, borrowed from [13, 25, 26]: 1. ϕ(s) = s + τ with τ ≥ 0. One has ϕ ∈ G and Cϕ is a (surjective) isometry if and only if τ = 0. 2. ϕ(s) = c0 s with c0 ≥ 1. Cϕ is an isometry (non-surjective if c0 ≥ 2). 3. ϕ(s) = c1 + c2 2−s with mandatorily c1 ≥ |c2 | + 1/2. When c1 = |c2 | + 1/2, Cϕ is not compact ([20, 27] independently). When c1 > |c2 | + 1/2, Cϕ is “very” compact! Indeed (anticipating on approximation numbers an (T ) to come) one can prove (see, e.g. [26]) that in that case an (Cϕ ) ≤ Cr n

where r =

2|c2 | < 1. 2c1 − 1

(8.6.9)

But in order to go deeper in the connections between operators and symbols on H2 , we must recall some facts on operators acting on a Hilbert space.

8.7 Compact Operators and Approximation Numbers

263

8.7 Compact Operators and Approximation Numbers 8.7.1 Definitions Let X be a Banach space, X ∗ its dual space, B X its closed unit ball, and L(X ) the Banach algebra of bounded operators T : X → X ; one defines the compactness of T ∈ L(X ) by requiring T (B X ) to be relatively compact in norm. Note in passing the following interesting result [3]: T (B X ) closed for all T ∈ L(X ) ⇔ X reflexive. Indeed, if X is reflexive, let y ∈ T (B X ) and (xn ) a sequence in B X such that w T (xn ) → y in norm, implying T (xn ) −→ y, where w indicates the weak converw gence. Taking a subsequence, we can also assume that xn −→ x with x ∈ B X . Hence, w T (xn ) −→ T (x), and y = T (x), via the uniqueness of the weak limit, showing that T (B X ) is closed (more conceptually, B X is weakly compact since X is reflexive, and T weakly continuous, so T (B X ) is weakly compact and hence closed). Conversely, if one tests the assumption against the rank one operator T defined by T (x) = f (x) e where f ∈ X ∗ and e ∈ X − {0}, one sees that every continuous linear form f on X attains its norm. It ensues by a result of James ([28], vol. 1, p. 315) that X is reflexive. This applies to Hilbert spaces H : T ∈ L(H ) will be compact if T (B H ), always closed, is moreover compact in norm. We now give a general compactness criterion for composition operators on a uc Banach space X of analytic functions on Ω. If f n , f ∈ H (Ω), we write f n −→ f to denote the uniform convergence of f n to f on compact subsets of Ω (mostly Ω = D or Ω = C1/2 ). If A ⊂ X ∗ is a complete subset of X ∗ , weak convergence can be tested as follows (this is classical and easy): w

xn −→ x ⇔ xn = O(1) and x ∗ (xn ) → x ∗ (x) for all x ∗ ∈ A. A useful corollary (take for A the set of point evaluations) is Lemma 8.7.1 If f n and f belong to the Banach space X , the following are equivalent: w 1. f n −→ f . uc

2. f n −→ f and f n = O(1). Let now ϕ : Ω → Ω an admissible symbol for a reflexive Banach space X ⊂ H (Ω), and Cϕ : X → X the associated composition operator, Cϕ f = f ◦ ϕ. A consequence of Lemma 8.7.1 is the following compactness criterion ([1], p. 29 for H 2 , but the proof works for all reflexive Banach spaces of analytic functions on an open set Ω of Cd , see also Exercise 6):

8 Composition Operators on the Space H2 of Dirichlet Series

264

Theorem 8.7.1 If ϕ : Ω → Ω is an admissible symbol for the reflexive Banach space X , the following are equivalent: 1. Cϕ : X → X is compact. uc

2. If f n −→ f with f n X = O(1), then f n ◦ ϕ → f ◦ ϕ in X . We next try to “quantify” the degree of compactness of a given operator T : X → X , through its approximation numbers which we define now. Let n be a positive integer. The n-th approximation number an (T ) of T is the distance of T to operators of rank < n ([29], Chap. 2), that is, an (T ) =

  sup T f − R f . inf

T − R = inf rank(R) 0.

|λn | p ≤

∞  n=1

anp

8.7 Compact Operators and Approximation Numbers

267

Proof We refer to ([29], p. 157) for a proof. Passing from the multiplicative form of Weyl’s inequalities to their (seemingly weaker) additive form is due to general properties of convex functions ([29], p. 157). Remark The tempting inequality |λn | ≤ an is wrong, as shown by an example of H. König in dimension n ([29], p. 154). This being said, the following substitute is almost as useful. Lemma 8.7.3 With the previous notations, one has for every integer q ≥ 2 q

1

|λqn | q−1 ≤ a1q−1 an . In particular |λ2n |2 ≤ a1 an . Proof In Lemma 8.7.2, change n into qn and raise to power q/(q − 1) to get 1  1  1



q |λqn | q−1 ≤ a1 · · · aqn (q−1)n ≤ a1n an(q−1)n (q−1)n = a1q−1 an .

We will be able to exploit those multiplicative Weyl’s inequalities in our context thanks to the following result, interesting in itself, due to Bayart [13].  −s ∈ G defining a compact composition Theorem 8.7.4 Let ϕ(s) = c0 s + ∞ n=1 cn n 2 operator on H . Then, the eigenvalues of Cϕ are simple and their value is (a) λn = [ϕ (α)]n−1 , n = 1, 2, . . . , when c0 = 0, where α is the fixed point of ϕ in C1/2 . (b) λn = n −c1 , n = 1, 2, . . . , when c0 = 1. (c) 1 if c0 ≥ 2, and then σ(Cϕ ) = {0, 1}. The assertion (a) of Theorem 8.7.4 requires a justification. Here it is. Lemma 8.7.4 Let Cϕ : H2 → H2 with c0 = 0. Then, ϕ has a fixed point in C1/2 . Proof We rely (in the case c0 = 0) on a mapping property of Proposition 8.7.1 which we recall: (8.7.5) ϕ(s) = s + iτ ⇒ ϕ(C1/2 ) ⊂ C1/2+ε for some ε > 0. Observe now that ϕ cannot be a vertical translation s + iτ since c0 = 0. Let T : D → C1/2 be the conformal map given by T (z) =

1 1−z + · 2 1+z

Let ω = T −1 ◦ ϕ ◦ T : D → D. Suppose that ω has no fixed point in D. Then, by the Denjoy–Wolff theorem ([1], p. 78), there exists a point u ∈ ∂D such that uc ωn −→ u, where ωn designates the n-th iterate of ω. If u = −1, T (u) = ∞,

8 Composition Operators on the Space H2 of Dirichlet Series

268 uc

and ϕn = T ◦ ωn ◦ T −1 −→ T (u) ∈ ∂C1/2 . But this is impossible by (8.7.5) since uc ϕn (C1/2 ) ⊂ C1/2+ε . If u = −1, one has ϕn −→ ∞. This is again impossible because, by Theorem 8.4.1, the Dirichlet series of ϕ converges uniformly on C1/2+ε so that there exists a bounded subset B of C1/2 , for example, B = ϕ(C1/2+ε ), such that ϕn (C1/2 ) ⊂ B for n ≥ 2; indeed, n ≥ 2 ⇒ ϕn (C1/2 ) ⊂ ϕ2 (C1/2 ) ⊂ B. All this shows by contradiction that ω possesses a fixed point u inside D and that ϕ possesses the fixed point α = T (u) ∈ C1/2 . In what follows, we present a study of singular numbers for composition operators on the Hardy–Dirichlet spaces H p , borrowed from [9, 21]. A change of scale when we pass from H p to H p (r n replaced by n −A ) will occur when c0 ≥ 1. We begin by the case p = 2.

8.8 A Lower Bound The aim of this section is to show the following result.  −s Theorem 8.8.1 Let ϕ(s) = c0 s + ψ(s) = c0 s + ∞ ∈ G, inducing a comn=1 cn n 2 pact composition operator on H . Then, one always has c1 > 0. Moreover, δ denoting a positive constant: • If c0 = 0, one has an (Cϕ ) ≥ δr n for some δ > 0 and some 0 < r < 1. / S p for p ≤ 1/c1 . • If c0 ≥ 1, one has an (Cϕ ) ≥ δ(n log n)−c1 . In particular, Cϕ ∈ The result of 1. is optimal, and that of 2. as well, up to the logarithmic factor. Proof If ψ(s) = c1 is constant, we get n −ϕ(s) H2 = n −c1 → 0 by compactness, hence the inequality c1 > 0. If ψ is not constant, Proposition 8.4.3 and the assumption ϕ : C0 → C0 show that ψ : C0 → C0 and then Lemma 8.3.4 shows that, again, c1 > 0 (we even get even c1 > 1/2 when c0 = 0). Let us next separate two cases. 1. c0 = 0. We will rely on a simple lemma ([9], Lemma 6.1) to reduce the situation to the case when ϕ has an attractive, but not super-attractive, fixed point. Lemma 8.8.1 Let α and β two points in C1/2 .There exists a Dirichlet polynomial χ ∈ G such that χ(β) = α and χ (β) = 0. Let then α ∈ C1/2 satisfying ϕ (α) = 0 and β = ϕ(α) ∈ C1/2 . We set ψ = χ ◦ ϕ ∈ G. This ψ satisfies ψ(α) = χ(β) = α and ψ  (α) = χ (β)ϕ (α) = 0.

8.8 A Lower Bound

269

Moreover, Cψ = Cϕ Cχ , and hence by the ideal property of approximation numbers, an (Cψ ) ≤ Cχ an (Cϕ ). Up to replacing ϕ by ψ, one can assume, without loss of generality, that ϕ(α) = α and ϕ (α) = 0. Let r0 := |ϕ (α)| > 0. One now uses Theorem 8.7.4 and Lemma 8.7.3 to get r04n−2 = |λ2n |2 ≤ a1 an , and the result ensues. 2. c0 ≥ 1. We shall give two proofs, quite different in spirit, the first one working only in the case c0 = 1. Proof A. We use Theorem 8.7.4 and the Weyl inequality of Lemma 8.7.3 with qn, where q ≥ 2 is an integer to adjust later. Let γ1 = c1 . Since λ j (Cϕ ) = j −c1 , Weyl’s inequality gives us q

q

1

(qn)−γ1 q−1 = |λqn | q−1 ≤ a1q−1 an  an . This implies γ1

an  (qn)−γ1 (qn)− q−1 . Adjusting q = qn as the integer part of log n + 2 and observing that (qn)1/q → e, we finally get an  (n log n)−γ1 , QED. Proof B. It is based on the definition of the an as Bernstein numbers and on a good choice of the n-dimensional test space E. One denotes by ( pk ) the sequence of prime numbers.  −s Lemma 8.8.2 Fix an integer n ≥ 1. Let ϕ(s) = c0 s + ∞ ∈ G with c0 ≥ j=1 c j j 2 1. Let also E be the n-dimensional  subspace of H generated by the unit vectors p1−s , p2−s , . . . , pn−s . For all f (s) = nk=1 bk pk−s in E, one has Cϕ f (s) = g(s) + h(s) where g(s) =

n 

bk pk−c1 ( pkc0 )−s

and h, g = 0.

k=1

(8.8.1) Proof The proof of Theorem 8.3.1 (see also [10]) showed us that the following formal computation is permitted for finding the Dirichlet coefficients of Cϕ ( pk−s ), 1 ≤ k ≤ n: ∞ ∞   (−c j log pk )l −ls  j 1+ Cϕ ( pk−s ) = pk−c0 s pk−c1 l! j=2 l=1 =: pk−c0 s pk−c1 (1 +

 m≥2

αk,m m −s ) =: gk (s) + h k (s)

8 Composition Operators on the Space H2 of Dirichlet Series

270

−c1 with pkc0 )−s , and it remains to observe that the two vectors g = n gk (s) = pk (  n k=1 bk gk and h = k=1 bk h k are orthogonal because their Dirichlet spectra are disjoint; in fact, the p j being prime and c0 being ≥ 1, an equality of the form pkc0 = p cj 0 m with m ≥ 2 is impossible, whether j is equal to k or not.

The second proof of 2. now works as follows, and is also valid for c0 ≥ 2 (whereas the spectral method of the first proof fails in that case). We choose E as in Lemma 8.8.2 and let f a vector of the unit sphere S E of E. Using the lemma, we get n  

 −2γ 1/2 |bk |2 pk 1

Cϕ ( f ) H2 = g + h H2 ≥ g H2 = k=1

≥

n



|bk |2 pn−2γ1

1/2

= pn−γ1 .

k=1 −γ

It ensues that bn (Cϕ ) ≥ γ E (Cϕ ) ≥ pn 1 . Now, as a consequence of one half of the prime number theorem under the form given by Chebyshev ([32], pp. 340–345), one has  x 1 π(x) := · log x p≤x Specializing to x = pn , for which π(x) = n, we deduce that pn  n log n. This gives the claimed lower bound. We could of course have used the full prime number theorem, but we wished to emphasize the elementary character of Theorem 8.8.1. Let us finally prove the optimality. 1. c0 = 0. One takes care of this case through the example ϕ(s) = c1 + c2 2−s with c1 > 1/2 + |c2 | for which [26] we saw that an (Cϕ ) ≤ δr n

with r =

2|c2 | < 1. 2c1 − 1

More generally [9], if ϕ ∈ G and if ϕ(C1/2 ) is a compact subset of C1/2 , an (Cϕ ) ≤ Cr n for some r < 1. But the proof is more complicated and uses the notion of Carleson measure. 2. c0 ≥ 1. Let ϕ(s) = c0 s + c1 . Then, an (Cϕ ) = n −γ1 . Write indeed ϕ= T ◦χ

with χ(s) = c0 s

and T (s) = s + c1 .

Then Cϕ = Cχ C T , Cχ is an isometry of H2 and left multiplication by an isometry preserves approximation numbers. Therefore, an (Cϕ ) = an (C T ) = n −γ1 for C T is diagonal with diagonal elements n −c1 on the canonical Hilbertian basis (u n ) of H2 , u n (s) = n −s .

8.9 Upper Bounds

271

8.9 Upper Bounds We begin by a general result [13, 20]. Theorem 8.9.1 Let ϕ ∈ G. Then: • If Cϕ is Hilbert–Schmidt, ϕ(C0 ) ⊂ C1/2 . • If ϕ(C0 ) ⊂ Cν with ν > 1/2, Cϕ is Hilbert–Schmidt. The proof of the second item consists in showing that ([13], see also Exercise 5):

n −ϕ 22 ≤ n −2ν . Upper bounds for the an (Cϕ ), ϕ ∈ G, are also given in [9] in terms of Carleson measures, and the results obtained are very sharp for “d-dimensional” (d ≥ 2) symbols of the form d  ϕ(s) = c1 + c j q −s j , j=1

where the q j ’s are independent integers (i.e. the numbers log q j are rationally independent in R, for example, q1 = 2, q2 = 6). In that case, by the independence of the q j and Kronecker’s theorem, the membership of ϕ to G amounts to K (ϕ) := c1 −

d 

|c j | ≥ 1/2.

(8.9.1)

j=1

When K (ϕ) > 1/2, Cϕ is compact and even Hilbert–Schmidt (see [27], this also follows from Theorem 8.9.1). An extension was obtained in ([23], Theorem 2) under the following form:  Theorem 8.9.2 Let ϕ(s) = c1 + dj=1 P j (q −s j ) ∈ G with the positive integers q j independent, and P j nonconstant polynomials. Then, Cϕ is compact if and only if ϕ(C0 ) ⊂ Cν for some ν > 1/2 or d ≥ 2. We are interested here in the critical case K (ϕ) =

1 · 2

The precise statement is [9].

 Theorem 8.9.3 Let ϕ(s) = c1 + dj=1 c j q −s j ∈ G with the positive integers q j independent, c j = 0 for 1 ≤ j ≤ d and K (ϕ) = 21 , thus inducing a bounded composition operator on H2 . Then :

• If 1 ≤ d < ∞, one has n −(d−1)/2  an (Cϕ )  n −(d−1)/2 (log n)(d−1)/2 .

8 Composition Operators on the Space H2 of Dirichlet Series

272

• If d = ∞, Cϕ belongs to

 p>0

Sp.

The result of 1. is optimal, and that of 2. as well, up to the logarithmic factor. The proof, fairly technical, uses among other things barycentres of reproducing kernels (this result was extended by Bayart and Brevig in [33], Theorem 16 and Corollary 17). One obtains, in particular, up to a logarithmic factor: an (Cϕ ) ≈ n −(d−1)/2 . In particular, one knows exactly which Schatten classes are allowed: Cϕ ∈ S p ⇔ p >

2 · d −1

/ S2 (a result already in [27]) but Cϕ ∈ S p for For d = 2, we therefore have Cϕ ∈ every p > 2, whereas in [27] the result is proved only for p ≥ 4. For d = 1, we recover a result of [13] or [27]: Cϕ is not compact. Remark The reader will find in [9] more details on upper bounds of the approximation numbers of operators Cϕ acting on H2 , where more elaborate techniques (Carleson measures, interpolation sequences, resolution of the ∂-bar equation with estimates) are used, as well as the definition of the an ’s as Gelfand numbers.

8.10 The Case of H p -spaces 8.10.1 Reminder For p > 0, let H p (C1/2 ) be the Hardy space of the half-plane C1/2 , namely, the set of functions h analytic in C1/2 for which  p

h H p (C1/2 ) := sup

∞

σ>1/2 −∞

|h(σ + it)| p dt < ∞.

(8.10.1)

Every h in H p (C1/2 ) has (Fatou’s theorem) a non-tangential limit at almost each point of the vertical line σ = 1/2, and the corresponding limit function t → h(1/2 + it) belongs to L p (R); the L p -norm of this function is the same as the H p -norm defined by (8.10.1). We refer to [12] for more details. We saw, in the case p = 2, the importance of local (or global) embedding (a consequence of the Montgomery–Vaughan inequality of Chap. 111) in the proof of the sufficiency in Theorem 8.2.1 when c0 = 0, this embedding is helping to make the connection between both spaces H 2 (C1/2 ) and H2 [22]. As we already said, the failure of this ingredient for the spaces H p , 0 < p < 2, recently proved by A. Harper, implies the failure of Theorem 8.2.1 for those spaces.

8.10 The Case of H p -spaces

273

8.10.2 Failure of Embedding for 0 < p < 2 We refer to Chap. 6, Sect. 6.5, for the definition of the Hardy spaces H p , 1 ≤ p ≤ ∞. Recall that those still all live on C1/2 , since they are contained in H1 and since, ∞ spaces −s for f (s) = n=1 an n ∈ H1 , Helson’s inequality ∞  |an |2 1/2 ≤ f 1 d(n) n=1

shows that σa ( f ) ≤ 1/2, by Cauchy–Schwarz. We saw at the beginning of this chapter that, for composition operators, there is no big difference between H p and H 2 , the reason being in short “Blaschke product”. More precisely, if f ∈ H p is zero-free, one writes it as f = g 2/ p with g ∈ H 2 , g 2 = f p and one is back to H 2 . And otherwise, f writes as f = B F where B is a Blaschke product (in particular, unimodular at the boundary) and F a zero-free function such that F p = f p and we are back again to H 2 . In the case of the “curly” H p -spaces, nothing as pleasant happens, and there are big differences between H p and H2 ! Even though, according to the Aleman–Olsen–Saksman theorem mentioned in Chap. 6, the “monomials” n −s still form a (conditional) Schauder basis of H p for 1 < p < ∞. We will encounter here a major difficulty: the membership of ϕ in G is necessary but not sufficient, in general, for ensuring that Cϕ : H p → H p , except when c0 ≥ 1 or when p is an even integer. And contrary to the case p = 2, a full description of the admissible symbols is not known in December 2020. We will come back to that in the final section, but begin by a positive result due to F. Bayart. One can enunciate a necessary and sufficient condition when c0 ≥ 1, but must content oneself with what follows when c0 = 0. Theorem 8.10.1 (Bayart) Let ϕ : C1/2 → C1/2 analytic and 1 ≤ p < ∞. (a) If Cϕ : H p → H p , one must have ϕ ∈ G. (b) Cϕ is a contraction on H p if and only if ϕ ∈ G and c0 ≥ 1. (c) If ϕ ∈ G and c0 = 0, Cϕ is bounded on H p when p is an even integer. For a detailed proof, we refer to the two seminal papers [13, 20], where the spaces H p are coined. We just mention here that if p = 2k is an even integer, the result is clear since f k ∈ H2 whenever f ∈ H p with equal norms:

f pp = f k 22 . As for (b), we use the norm of the pointwise evaluation on H p at a ∈ C1/2 and the mapping equation (see [13]), namely,

1/ p and Cϕ∗ (δa ) = δϕ(a) .

δa = ζ(2a)

8 Composition Operators on the Space H2 of Dirichlet Series

274

We thus get, in case c0 = 0, quite analogously to the case p = 2, and assuming that Cϕ is bounded on H p :

1/ p

Cϕ ≥ ζ(2c1 ) > 1. Remark By analogy with an old result of Hardy and Littlewood (the so-called majorant property, see [34]), we conjecture that the case of even integers is the only one for which the continuity of Cϕ : H p → H p when c0 = 0 is ensured by the membership of ϕ in the Gordon–Hedenmalm class. Let us switch to negative facts. Here is a deep result of Harper [35, 36]: Theorem 8.10.2 (Harper) Let 0 < p < 2. Then, for every A > 0, there exists f ∈ H p with “H p -norm” equal to one such that 

1 0

 1  p  + it  dt > A. f 2

As a consequence of Theorem 8.10.2, we actually have: Theorem 8.10.3 Let 1 ≤ p < 2. Then, there exists a symbol ϕ ∈ G, satisfying c0 = c0 (ϕ) = 0, such that Cϕ is not bounded on H p . Proof We simply take ϕ(s) =

1 1 − 2−s + · 2 1 + 2−s

(8.10.2)

Suppose that Cϕ : H p → H p . Denote the Bohr lift by . The isometric identification of H p with H p (T∞ ) shows that, for f ∈ H p , with ( f ◦ ϕ)(z) = f

1 2

+

1 − z1  , z = (z 1 , . . . , z n , . . . , ) ∈ T∞ 1 + z1

one has, abbreviating . H p into . p :

f ◦ ϕ pp =

1 2π

1 = π



 π  1 1 − eiu  p u  p 1   1  f + − i tan  du du =  f  iu 2 1+e 2π −π 2 2 −π π

 1    p dt  p  1  1 f + it  + it   dt. f  2 1 + t2 2 R 0

Now, by Harper’s result, we can find f ∈ H p with norm one, such that  1  1  p  dt is large as we wish. This shows by contradiction that Cϕ can 0 f 2 + it not be bounded on H p .

8.10 The Case of H p -spaces

275

8.10.3 Positive Results We will proceed with a more optimistic note (see [21] for more details); when Cϕ : H p → H p with p = 2, ∞ is bounded, even though the dual space of H p is poorly known, we can go beyond the Hilbertian situation, and say many interesting things; some proofs are more difficult, and the results not so precise, while being very similar. We sketch a proof of those facts. First the description of the point spectrum of Cϕ , given by Theorem 8.7.4, extends to the H p -case [21]; we recall it.  −s Theorem 8.10.4 Let ϕ(s) = c0 s + ∞ n=1 cn n , defining a compact composition p operator on H . Then, the eigenvalues of Cϕ have multiplicity one and their value is (a) λn = [ϕ (α)]n−1 , n = 1, 2, . . . , when c0 = 0, where α is the fixed point of ϕ in C1/2 . (b) λn = n −c1 , n = 1, 2, . . . , when c0 = 1. (c) 1 when c0 ≥ 2, and in that case σ(Cϕ ) = {0, 1}. This description of the spectrum will be exploited with help of the following extension by Pietsch [37] of Weyl’s inequalities to a Banach space. Those inequalities must be weakened, under a form which will meet our needs. Theorem 8.10.5 (Pietsch) Let T : X → X be a compact operator of a Banach space X to itself. Then, for every integer n ≥ 1: |λ2n (T )| ≤ e

n 

a j (T )

1/n

,

(8.10.3)

j=1

where λ j (T ) and a j (T ) denote, respectively, the eigenvalues and the approximation numbers of T , in decreasing order. We will rely on those theorems to sketch the proof of the following theorem [21]. Theorem 8.10.6 that c0 is a non-negative integer, p ≥ 1, and also that  Assume −s c n = c0 s + ψ(s) ∈ G is nonconstant and generates a comϕ(s) = c0 s + ∞ n=1 n pact composition operator Cϕ on H p . Then, c1 =: γ1 > 0. Moreover, (a) If c0 = 0, an (Cϕ )  r n for some 0 < r < 1. (b) If c0 ≥ 1, an (Cϕ ) ≥ δ p (n log n)−γ1 ,where δ p > 0 only depends on p. In particular, ∞  [an (Cϕ )]1/γ1 = ∞. (8.10.4) n=1

Those lower bounds are optimal in general.

276

8 Composition Operators on the Space H2 of Dirichlet Series

Proof The inequality γ1 > 0 is shown as in the Hilbertian case p = 2, separating the cases c0 = 0 and c0 ≥ 1. We next limit ourselves to the case p > 1 and begin by the optimality issue. Let first ϕ ∈ G with c0 = 0, and hence ϕ(C0 ) ⊂ C1/2 . Assume that, moreover, ϕ(C0 ) is compact. One then shows ([21], Theorem 8.1(a)) that there exists 0 < r < 1 such that an (Cϕ )  r n . Let next T A (s) = s + A with A > 0. Using that the u n , u n (s) = n −s , form a Schauder basis of H p for p > 1 [38], we will show that an (C TA )  n −A ([21], Theorem 8.1(b)). Indeed, denote . the norm on H p and let R be the operator of rank n − 1 defined by n−1 ∞   Rf = j −A x j u j for f = xju j. j=1

j=1

Using the contraction principle for Schauder bases [28], we easily see that     −A  C T ( f ) − R( f ) =  j x j u j  ≤ 2n −A f

A j≥n

so that

an (C TA ) ≤ C TA − R ≤ 2n −A .

This extends, using the ideal property of approximation numbers, to the general case ϕ(s) = c0 s + ψ(s) satisfying ϕ(C0 ) ⊂ C A , by writing ϕ A (s) = ϕ(s) − A, T A (s) = s + A and then ϕ = T A ◦ ϕ A , an (Cϕ ) ≤ Cϕ A an (C TA )  n −A . We are left with proving the lower bounds in the general situation. Let us separate two cases. Case c0 = 0. The proof is similar to the first proof of the case p = 2, using this time Theorems 8.10.4 and 8.10.5, the “Tauberian” argument being the same. Case c0 ≥ 1. One must adapt Lemma 8.8.2 under the form ([21], Lemma 7.1):  −s Lemma 8.10.1 Let n ≥ 1 be an integer. Assume that ϕ(s) = c0 s + ∞ j=1 c j j p belongs to G with c0 ≥ 1. Let E be the n-dimensional subspace of H generated by the vectors p1−s , p2−s , . . . , pn−s .Then, for all functions f (s) = nk=1 bk pk−s ∈ E, it holds: Cϕ f (s) = g(s) + h(s) where g(s) =

n 

bk pk−c1 ( pkc0 )−s and g H p ≤ Cϕ f H p .

k=1

(8.10.5)

8.11 A Few Updates and Remarks

277

One can now end the proof of Theorem 8.10.6. Choose E as in Lemma 8.10.1 and let f ∈ H p be a unit vector of E. Lemma 8.10.1 and a Bohr lift (replacement of g by g with the notations of Chap. 1) give n       ck pk−c1 z kc0 

Cϕ ( f ) H p ≥ g H p =  k=1

H p (T∞ )

n     = ck pk−c1 z k  k=1

H p (T∞ )

the last relation following from the invariance of the Haar measure m ∞ of T∞ under the map (z j ) → (z cj0 ). Applying twice the Khintchine inequalities to the Steinhaus variables z j , we get

Cϕ ( f )

Hp

n      ck pk−c1 z k  k=1

H 2 (T∞ )

n      pn−c1  ck z k  k=1

As a consequence,

≥

H p (T∞ )

n 

 pn−c1 

  ck z k 

k=1

H 2 (T∞ )

= pn−c1 f H p .

an (Cϕ ) ≥ bn (Cϕ )  pn−c1 .

By the prime number theorem, used here unscrupulously, we know that pn  n log n, and this ends the proof. One might also use (see [21]) Theorems 8.10.4 and 8.10.5.

8.11 A Few Updates and Remarks We end up this chapter with quite recent results. 1. The failure of the local embedding for 0 < p < 2 seems once more related to the non-complementation of H p (T∞ ) in L p (T∞ ), which prevents to see H p as a complex interpolate between H2k and H2(k+1) when 2k < p < 2(k + 1), k being a non-negative integer. For a more specific statement in this regard, see the quite recent paper by Bayart and Mastylo [39], which formulates the obstructions to interpolation in a very precise way. 2. Problem 3 in the survey [40] asks about the Dirichlet series analogue of the sharp upper bound obtained from Littlewood’s subordination principle in the disc (Theorem 8.1.1), mainly for p = 2. Specifically, how large can the norm of Cϕ be on H2 in terms of ϕ(+∞) = c1 ∈ C1/2 ? When one has 0 < (c1 ) − 1/2 ≤ 1, it does not require much additional work to conclude that the sharp upper bound is 

Cϕ ≤

2 (c1 ) − 1/2

(8.11.1)

8 Composition Operators on the Space H2 of Dirichlet Series

278

following the paper by Brevig [41]. Here (c1 ) − 1/2 = (ϕ(+∞)) − 1/2 is the distance the boundary of C1/2 in the same way 1 − |ϕ(0)| is the distance to the boundary of D. The relation (8.11.1) can be proved in two steps. First step: for b > 0, let Pb (t) = π(t 2b+b2 ) be the Poisson kernel, and Cb denote the optimal constant in the global embedding  R

| f (1/2 + it)|2 Pb (t) dt ≤ Cb f 2H2 ,

(8.11.2)

from Eq. (8.5.3) in Theorem 8.5.3. With x > 0, we get 

 R

x it Pb (t) dt =

eitb log x R

dt = e−b| log x| + 1)

π(t 2

by Fourier transform of the function t → (1 + t 2 )−1 . If f (s) =  the well-known −s we get with x = n/m that n≥1 an n  R

| f (1/2 + it)|2 Pb (t) dt =

∞ ∞   am a n (mn)b−1/2 am an . √ e−b| log(n/m)| = [max(m, n)]2b mn m,n=1 m,m=1

When 0 < b ≤ 1 it is easy to compute the optimal constant Cb = 2/b following the first few pages of Chap. IX in [42]. Indeed, the case b = 1/2 is the first exercise of this chapter and the case 0 < b ≤ 1 has precisely the same proof. After the Schur test or the Cauchy–Schwarz inequality, the upper bound boils down to estimating m −b

m  n=1

n b−1 + m b

∞ 

n −b−1 ≤ m −b

n=m+1

 m 0

x b−1 d x + m b

 ∞ m

x −b−1 d x =

1 1 + · b b

When b > 1 it is more complicated, since the first sum cannot be estimated in the same way with an integral (see [41]). But we can easily get Cb ≤ 1 + b−1 for b > 1. For the lower bound, we simply set an = n −1/2−ε and let ε → 0+ which implies that Cb ≥ 2/b holds for all 0 < b < ∞ (see Lemma 7 in [41]). In [41], the conclusion Cb = 2/b is extended up to b = 1.48 . . . and with some effort one can push it to b = 1.5. In general, the problem to determine the optimal Cb is open and probably quite difficult. Note that trivially Cb ≥ 1, and as was shown in (8.6.8), we have Cb ≥ ζ(1 + 2b). Second step : In the proof of sufficiency for c0 = 0 in Sect. 6.2.2, we set c1 = ϕ(+∞) and b = (c1 ) − 1/2 and τ = (c1 ). We also set f τ (s) = f (s + iτ ) and recall that

f H2 = f τ H2 for any τ ∈ R. Following the proof line for line, we get 

f ◦ ϕ 2H2 ≤ | f τ (1/2 + it)|2 Pb (t) dt. R

8.12 Exercises

279

√ Hence, we get the upper bound Cϕ ≤ Cb where Cb is the optimal constant from Eq. (8.5.3) and b = (c1 ) − 1/2. It is easy to see that this estimate is sharp for any c1 ∈ C by considering ϕ(s) =

1 1 − 2−s + iτ + b 2 1 + 2−s

and arguing as the proof of Theorem 8.10.3. Combining everything, we find that the upper bound (8.11.1) holds for 0 < (c1 ) − 1/2 ≤ 1. 3. The example of Theorem 8.10.3 was shown orally to us by Perfekt [43]. It is in fact the “new” part of Theorem 3 in the recently published paper [33]. As the authors mention, their work dates back to 2016 and actually precedes Harper’s counterexample. Their key observation is that the local embedding for H p is equivalent to the boundedness of every composition operator Cϕ on H p for ϕ ∈ G. By the power trick and the fact that ζ(s + 1/2 + ε) does not vanish in C1/2 , they get that the sharp upper bound   1p 2

Cϕ ≤ (c1 ) − 1/2 holds also for p = 2k when 0 < (c1 ) − 1/2 ≤ 1. 4. The characterization of compactness for composition operators Cϕ ’ when ϕ belongs to the Gordon-Hedenmalm class was recently achieved by O. F. Brevig and K. M. Perfekt when c0 = 0, in terms of a new counting function which replaces the Nevanlinna counting function.

8.12 Exercises 1. For 0 < a < 1, let ϕ(z) = az + 1 − a : D → D. By properly using the Schur test and the mapping equation for H 2 , show that

Cϕ = a −1/2 . Show more precisely that Cϕ e = a −1/2 where . e designates the essential norm, or the distance to compact operators (Hint: the norm of Cϕ is approximated on a weakly null sequence of unit vectors).   −s −s and G(s) = ∞ be two convergent Dirichlet 2. Let F(s) = ∞ n=1 f n n n=1 gn n series. Show that σu (F G) ≤ max(σa (F), σu (G)). In other terms, the Dirichlet product of an absolutely convergent and a uniformly convergent series is uniformly convergent. 3. Detail the proof of the orthogonality of the random multiplicative functions χ(n), i.e.

280

8 Composition Operators on the Space H2 of Dirichlet Series

E[χ(m)χ(n)] = 0

whenever m = n.

4. Detail the proof of the second equivalence (that is, the implication from left to right!) at the beginning of Sect. 4.5 (the first equivalence is the well-known Khintchine–Kolmogorov theorem). 5. Let ϕ ∈ G with ϕ(C0 ) ⊂ Cν and ν > 1/2. For ε > 0, let ϕε (s) = ϕ(s + ε). We abbreviate the norm in H2 by . . (a) Using Theorem 8.5.1, show that n −ϕε ≤ n −ν . (b) Show that n −ϕ ≤ n −ν , that Cϕ is Hilbert–Schmidt, and even that Cϕ ∈ S p , the p-Schatten class, for p > 1/ν. 6. (a) In the Banach space framework, why does one assume reflexivity in Theorem 8.7.1 ? (Hint: consider the identity from 1 to itself and remember ([28], vol. 1, p. 65) that 1 has the Schur property). (b) Notwithstanding, show that the criterion of Theorem 8.7.1 works for composition operators on X = H ∞ (Ω).

References 1. J. Shapiro, Composition Operators and Classical Function Theory (Springer, 1993) 2. D. Li, H. Queffélec, L. Rodriguez-Piazza, On approximation numbers of composition operators. J. Approx. Theory. 164, 431–459 (2012) 3. D. Li, H. Queffélec, L. Rodríguez-Piazza, A spectral radius type formula for approximation numbers of composition operators. J. Funct. Anal. 268(2), 4753–4774 (2015) 4. D. Li, H. Queffélec, L. Rodríguez-Piazza, Approximation numbers of composition operators on H p . Concr. Oper. 2, 98–109 (2015) 5. D. Li, H. Queffélec, L. Rodríguez-Piazza, Pluricapacity and approximation numbers of composition operators. J. Math. Anal. Appl. 474(2), 1576–1600 (2019) 6. D. Li, H. Queffélec, L. Rodríguez-Piazza, Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk. Trans. Amer. Math. Soc. 372(4), 2631–2654 (2019) 7. D. Li, H. Queffélec, L. Rodríguez-Piazza, Approximation numbers of composition operators on the Hardy space of the infinite polydisk. Integr. Equ. Oper. Theory 89(4), 493–505 (2017) 8. A. Defant, D. Garcia, M. Maestre, P. Sevilla-Peris, Dirichlet Series and Holomorphic Functions in High Dimensions. New Mathematical Monographs, vol. 37 (Cambridge University Press, 2019) 9. H. Queffélec, K. Seip, Approximation numbers of composition operators on the H 2 space of Dirichlet series. J. Funct. Anal. 268(6), 1612–1648 (2015) 10. J. Gordon, H. Hedenmalm, The composition operators on the space of Dirichlet series with square-summable coefficients. Michigan Math. J. 46, 313–329 (1999) 11. H. Hedenmalm, P. Lindqvist, K. Seip, A Hilbert space of Dirichlet series and a system of dilated functions in L 2 (0, 1). Duke Math. J. 86, 1–36 (1997) 12. J. Garnett, Bounded Analytic Functions (Springer, 2007) 13. F. Bayart, Compact composition operators on a Hilbert space of Dirichlet series. Illinois. J. Math. 47, 725–743 (2003) 14. B. Kashin, A. Saakyan, Orthogonal series. Transl. Math. Monogr. Am. Math. Soc. 75, (1989) 15. A. Olevskii, Fourier Series with Respect to General Orthogonal Systems, vol. 86 (Springer Verlag Ergebnisse der Mathematik, 1975)

References

281

16. K. Tandori, Über die orthogonalen Funktionen. X: Unbedingte Konvergenz. Acta Sci. Math. (Szeged) 23, 185–221 (1962) 17. J.P. Kahane, Some Random Series of Functions, 2nd edn. (Cambridge University Press, 1985) 18. H. Hedenmalm, E. Saksman, Carleson’s convergence theorem for Dirichlet series. Pacific. J. Math. 208, 85–109 (2003) 19. C. Cowen, B. McCluer, Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics (CRC Press, 1995) 20. F. Bayart, Hardy spaces of Dirichlet series and their composition operators. Monat. Math. 136, 203–226 (2002) 21. F. Bayart, H. Queffélec, K. Seip, Approximation numbers of composition operators on H p spaces of Dirichlet series. Ann. Inst. Fourier 66(2), 521–558 (2016) 22. J.F. Olsen, K. Seip, Local interpolation in Hilbert spaces of Dirichlet series. Proc. Am. Math. Soc. 136, 203–212 (2008) 23. M. Bailleul, O.F. Brevig, Composition operators on Bohr-Bergman spaces of Dirichlet series. Ann. Acad. Sci. Fenn. Math 41(1), 129–142 (2016) 24. M. Bailleul, P. Lefèvre, Some Banach spaces of Dirichlet series. Studia Math. 226(No1), 17–55 (2015) 25. C. Pommerenke, Boundary Behaviour of Conformal Maps (Springer, 2010) 26. P. Muthukumar, S. Ponnusamy, H. Queffélec, Estimate for the norm of a composition operator on the Hardy-Dirichlet space, Integr. Equ. Oper. Theory 90(1), 12 (2018). Art. 11 27. C. Finet, H. Queffélec, A. Volberg, Compactness of composition operators on a Hilbert space of Dirichlet series. J. Funct. Anal. 211, 271–287 (2004) 28. D. Li, H. Queffélec, Introduction to Banach spaces: analysis and probability. Translation of the French edition, with appendices by O. Guédon, G. Godefroy, G. Pisier, L. Rodríguez-Piazza, vol. 1 and 2, 700 pages (Cambridge University Press, 2018) 29. B. Carl, I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1990) 30. G. Lechner, D. Li, H. Queffélec, L. Rodríguez-Piazza, Approximation numbers of weighted composition operators. J. Funct. Anal. 274(7), 1928–1958 (2018) 31. A.V. Megretskii, V.V. Peller, S.R. Treil, The inverse spectral problem for self-adjoint Hankel operators. Acta Math. 174, 241–309 (1995) 32. G.H. Hardy, M. Wright, An Introduction to the Theory of Numbers (Clarendon Press, 1979) 33. F. Bayart, O. Brevig, Composition operators and embedding theorems for some function spaces of Dirichlet series. M. Zeit. 293(3–4), 989–1014 (2019) 34. G.H. Hardy, J.E. Littlewood, A problem concerning majorants of Fourier series. Quart. J. Math. Oxf. 6, 304–315 (1935) 35. A.J. Harper, Moments of random multiplicative functions, I: low moments, better than squareroot cancellation, and critical multiplicative chaos. arXiv:1703.06654 36. A.J. Harper, Moments of random multiplicative functions, I: low moments, better than squareroot cancellation, and critical multiplicative chaos. Forum of Mathematics, Pi (in Press) 37. A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces. Math. Ann. 247, 149–168 (1980) 38. A. Aleman, J.F. Olsen, E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series. arXiv: 1210.4292v1 [math.CV], 16 Oct 2012 39. F. Bayart, M. Mastylo, Interpolation of Hardy spaces of Dirichlet series. J. Funct. Anal. 277(3), 786–805 (2019) 40. H. Hedenmalm, Dirichlet Series and Functional Analysis. The legacy of Niels Henrik Abel. (Springer, Berlin, 2004), pp. 673–684 41. O. Brevig, Sharp norm estimates for composition operators and Hilbert-type inequalities. Bull. Lond. Math. Soc. 49(6), 965–978 (2017) 42. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, 2nd edn., Dover Phenix Editions (2005) 43. K.M. Perfekt, Private Communication (2018) 44. O. F. Brevig and K. M. Perfekt, A mean counting function for Dirichletseries and compact composition operators, (2020)

Index

Symbols (x,  z  y), 140, 188 λ N , 217 A(), 5 A P(R), 25 A N , 97 A∗N , 97 B 2 (U ), 207 C00 (G), 11 C0 (G), 3 D, 238 D0 , 238 G , 239 H ∗ (), 215 H ⊥ , 23 K t , 227 L N , 203 M(G), 3 P + (n), 136 P − (n), 188 S( N ), 139 S(x, y), 140 S p (), 187 T , 137 U N , 97 X ∗ , X ∗∗ , …, 21 Cθ , 95  f , 118 , 218 , 203 (n), 135, 188 , 17 x, 101 Var (g), 105 λ ∗ μ, 3 E(M), 132

H2 , 158 H∞ , 149 H p , 145, 167 P , 12 P N , 137

μ → μ, ˜ 3 ω(n), 135, 189 G, 24 σa , 97 σc , 97 σh , 99 σu , 97 εn , 9  f (γ), 4 ζ N , 203 d(n), 171 d A, 207 f ∗ g, 3 m, 2 Bad, 73 Wiener algebra, 5 π(y), 137 L 1 (G) = L 1 (G, m), 3 Pq,b , 210 Ta , 1 x → x, ˜ 9

A Aleman, 167 Almost-periodic, 25, 116 Approximation number, 264 Arithmetic functions, 113 Automatic, 110

© Hindustan Book Agency 2020 and Springer Nature Singapore Pte Ltd 2020 H. Queffelec and M. Queffelec, Diophantine Approximation and Dirichlet Series, Texts and Readings in Mathematics 80, https://doi.org/10.1007/978-981-15-9351-2

283

284 B Bagchi, 204, 232, 234 Baire category theorem, 116, 204 Baire space, 252 Baire’s theorem, 254 Balasubramanian, 149, 192 Balazard, 189 Banach algebra, 3, 148, 155 Banach space, 114 Banach–Steinhaus theorem, 114, 124 Bayart, 154, 187, 205, 267 Bayart and Mastylo, 277 Bayart, F., 273 Bergman space, 168, 205, 207 Berndtsson, 155 Bernstein, 265 Bernstein inequality, 130 Bernstein numbers, 264, 269 Beurling, 146 Beurling-Hörmander, 28 Birkhoff theorem, 41 theorem for flows, 48, 50 Birkhoff–Khinchin theorem, 205, 227 Birkhoff-Oxtoby theorem, 166 Blei, 142, 181 Bochner–Weil–Raikov, 9 Bohnenblust-Hille, 129, 139, 180 Bohr, 129, 204 compactification, 24 constant, 137 inequalities, 117, 194 point of view, 1, 129, 136 radius, 187 theorem, 151 Bohr and Jessen, 121 Bohr–Cahen formulas, 97 Bonami, 169 Borel–Bernstein theorem, 86 Borel–Cantelli’s lemma, 109 Borel measures, 3 Bounded variation, 102 Brevig, 278

C Calado, 149, 192 Cantor group, 8 Carathéodory domain, 209 Carleson, 155 Carleson measure, 270 Carleson’s theorem, 252 Carlson, 221

Index Carlson identity, 153, 163, 227 Cassels, 234 Cauchy-Davenport theorem, 36 Chang, 155 Character, 102, 245 Chebotarev lemma, 34 Closed graph theorem, 164 Coboundary, 39, 105 Compactness, 263 Composition operator, 161, 237 Continued fraction expansion, 74, 105 Convergent Dirichlet series, 147 Convergents, 75 Convolution, 3 Critical strip, 214

D Davenport, 234 de la Bretèche, 139 Denjoy–Koksma, 105 Denjoy–Wolff theorem, 267 Dickman function, 140 Dilated function, 145 Diophantine number, 74, 102 Dirichlet character, 197 polynomials, 125, 135 series, 1 Dirichlet polynomial, 259 Distinguished boundary, 131 maximum principle, 117, 121, 131 Divisible group, 12 Drobot, 205 Dual group, 4

E Embedding theorem, 160 Ergodic transformation, 52 Euler product, 114, 198 totient function, 212 Euler–Maclaurin, 124 Euler-MacLaurin summation formula, 178 Exponential type, 210

F Factorial, 113 Fatou’s theorem, 256 Fefferman, 167 Fejér, 162

Index Fibonacci, 125 Finite order, 221 Fourier transform, 4 Frerick, 180 Frontier, 245 Functional equations, 198

G Gallagher, 21 Gamma function, 218 Gauss dynamical system, 80 transformation, 72 Gaussian processes, 136 sum, 198 Gelfand numbers, 272 Gelfand topology, 5 General Dirichlet series, 96 Global embedding, 256 Gonek, 234 Gordon, 161 Gordon and Hedenmalm, 238 Gordon–Hedenmalm class, 239

H Haar measure, 2, 172, 217 Hahn–Banach theorem, 173, 206 Hankel forms, 176 Hankel operators, 266 Hardy, 27 Hardy and Littlewood, 106, 274 Hardy and Riesz, 96 Hardy-Dirichlet space, 165 Hardy-Hilbert space, 160 Hardy’s inequality, 168 Hardy space of the half-plane, 272 Harmonic, 245 Harnack’s inequality, 246 Harper, 176, 256, 274 Harper, A., 256, 272 Harper’s counterexample, 279 Hausdorff topology, 1 Hecke series, 101 Hedenmalm, 159, 161, 162 Heilbronn, 234 Helson, 169, 170, 173, 176 Helson forms, 171, 173 Helson matrices, 171 Herglotz theorem, 245 Herman, 105

285 Hermite-Padé approximant, 90 Hewitt and Williamson, 119 Hilbert inequality, 19, 177, 202 Hilbert-Schmidt, 172, 173 Hilbert–Schmidt class, 266 Holomorphy abscissa, 99 Hurwitz zeta function, 101 theorem, 204 zeta function, 101, 200, 234 Hyperbola method, 116 Hypercontractivity, 169

I Ideal property, 265 Improved Montel principle, 154 Independent set, 46 Interpolating sequences, 155 Interpolation sequences, 272 Invariant measure, 41 Inversion theorem, 13 Involution, 3 Irrationality exponent, 74

J Jensen’s lemma, 95 Jordan domain, 208, 210

K Kahane, 134, 252 Kernel, 176 Khintchine inequalities, 135 theorem, 93 Khintchine inequalities, 277 Khintchine–Kolmogorov theorem, 252 König, H., 267 Konyagin, 139 Kronecker flow, 48, 166, 227 lemma, 98 set, 62 theorem, 45, 49, 117, 205, 217, 227, 234 Kuzmin–Landau, 200

L Landau–Schnee theorem, 123 Lehmer, 56 Lerch functions, 232 Lévy theorem, 86 L-function, 102, 197, 214

286 Lifshits, 137 Lin, 155 Lindelöf, 116 Lindqvist, 162 Liouville number, 73 Littlewood–Paley type, 254 Littlewood’s subordination principle, 238 Local embedding, 176, 178, 256 Locally compact, abelian, group, 1 Lower density, 203

M Marcus, 136 Markov–Chebyshev, 228 Markov inequality, 211 Matheron, 187, 205 Maurizi, 116, 178 Maximal function, 132 Measure preserving transformation, 38 Mellin’s inversion formula, 219 Menchoff, 251 Mergelyan theorem, 208 Möbius function, 114, 124 Montel’s theorem, 165 Montgomery-Vaughan, 20 Montgomery-Vaughan generalized Hilbert’s inequality, 177 Multiplicative Dirichlet kernel, 176 Multiplicative semi-group, 163 Multipliers, 161, 164

N Nehari’s theorem, 171, 177 Neumann, 148

O Olsen, 160, 167 Open mapping theorem, 174, 177 Operator of translation, 1 Ortega-Cerdà, 174, 180 Ortega-Cerdà and Seip, 177 Ounaies, 180 Oxtoby theorem, 47 theorem for flows, 48, 166

P Padé approximant, 77 Paley–Wiener theorem, 214 ∂-bar equation, 272

Index Perfekt, 279 Perron–Landau, 98, 152, 201 Peter-Weyl theorem, 11 Pietsch, 275 Pisier, 136 Pisier norm, 135 Pisot-Vijayaraghavan number, 55 Plancherel theorem, 17 Point spectrum, 275 Poisson kernel, 169, 206, 246, 256, 278 Polylogarithmic function, 102 Pontryagin’s theorem, 23 Prime number theorem, 212 Principal character, 197 Products of Dirichlet series, 113

Q Quasi-sure, 253 Quasi-surely, 252 Queffélec, 139, 149, 178, 192

R Rademacher, 9 Rademacher variable, 134 Ramaré, 200 Random multiplicative character, 251 Random variables, 206 Rankin’s problem, 139 Reflexive, 263 Reich, 231 Reproducing kernel, 158, 237 Riemann, 205 Riemann hypothesis, 203 Riesz basis, 146 Riesz representation theorem, 10 Roth theorem, 70, 109 Rudin–Shapiro, 110 Rudin-Shapiro sequence, 138

S Saksman, 151, 162, 167 Saksman, E., 162 Salem, 134 Salem number, 56 Schatten classes, 238, 266 Schauder basis, 167, 276 Schmidt decomposition, 264 Schur matrix, 32 Schur property, 280 Schur test, 173, 175, 278 Seip, 156, 160, 162, 174, 180

Index Semi-group, 171 Sidon constant, 139, 178 Singular numbers, 238, 264 Slépian, 136 Spectrum, 275 Square-free, 138 Steinhaus, 143 Steinhaus theorem, 34 Steinhaus variables, 251, 277 Steinitz, 206 Steuding, 232 Stirling formula, 218 Stone-Weierstrass, 5 Strong mixing, 82 Subgaussian, 133 Sub-multiplicative, 137 Symbol, 237

T Tandori, 251 Tao, 31 Tchebychev, 137 Tenenbaum, 106 Three lines theorem, 249 Thue–Morse, 110 Thue–Morse sequence, 100 Tietze-Urysohn theorem, 11 Transitively, 34

287 U Uncertainty principle, 27 Uniformly distributed sequence, 54 Uniquely ergodic transformation, 38, 52 Upper density, 203

V Von Neumann theorem, 38 theorem for flows, 44 Voronin, 204, 233 Vukotic, 168

W Walfisz, 106 Walsh group, 8 Weak characters, 4 Weakly compact, 263 Weber, 137 Weierstrass theorem, 104, 208 Weyl, H., 266 Weyl’s inequalities, 275 Wiener’s lemma, 120

Z Zeta function, 96 Zygmund, 134