We present several formulae for the large t asymptotics of the Riemann zeta function ζ(s), s = σ + it, 0 ≤ σ ≤ 1, t >
158 40 1MB
English Pages 127 [128] Year 2022
Table of contents :
Cover
Title page
Part 1. Asymptotics to all Orders of the Riemann Zeta Function
Chapter 1. Introduction
1.1. The large 𝑡 asymptotics of 𝜁(𝑠) valid to all orders
1.2. The explicit form of certain sums
1.3. The explicit form of the difference of certain sums
1.4. Asymptotics of a twoparameter generalization of Riemann’s zeta function
1.5. Fourier coefficients of the product of two Hurwitz zeta functions
1.6. Several representations for the basic sum
Chapter 2. An Exact Representation for 𝜁(𝑠)
Chapter 3. The Asymptotics of the Riemann Zeta Function for 𝑡≤𝜂
Number 1351
On the Asymptotics to all Orders of the Riemann Zeta Function and of a TwoParameter Generalization of the Riemann Zeta Function Athanassios S. Fokas Jonatan Lenells
January 2022 • Volume 275 • Number 1351 (fifth of 6 numbers)
Number 1351
On the Asymptotics to all Orders of the Riemann Zeta Function and of a TwoParameter Generalization of the Riemann Zeta Function Athanassios S. Fokas Jonatan Lenells
January 2022 • Volume 275 • Number 1351 (fifth of 6 numbers)
Library of Congress CataloginginPublication Data CataloginginPublication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/memo/1351
Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2022 subscription begins with volume 275 and consists of six mailings, each containing one or more numbers. Subscription prices for 2022 are as follows: for paper delivery, US$1085 list, US$868 institutional member; for electronic delivery, US$955 list, US$764 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, add US$22 for delivery within the United States; US$85 for outside the United States. Subscription renewals are subject to late fees. See www.ams.org/helpfaq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/backvols. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 022845904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 029042213 USA. Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower righthand corner of the ﬁrst page of each article within proceedings volumes.
Memoirs of the American Mathematical Society (ISSN 00659266 (print); 19476221 (online)) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 029042213 USA. Periodicals postage paid at Providence, RI. Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 029042213 USA. c 2022 by the American Mathematical Society. All rights reserved. This publication is indexed in Mathematical Reviews , Zentralblatt MATH, Science Citation Index , Science Citation IndexTMExpanded, ISI Alerting ServicesSM, SciSearch , Research Alert , CompuMath Citation Index , Current Contents /Physical, Chemical & Earth Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acidfree and falls within the guidelines
established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1
27 26 25 24 23 22 22
Dedicated to Trevor Stuart with deep gratitude
Contents Part 1.
Asymptotics to all Orders of the Riemann Zeta Function
Chapter 1. Introduction 1.1. The large t asymptotics of ζ(s) valid to all orders 1.2. The explicit form of certain sums 1.3. The explicit form of the diﬀerence of certain sums 1.4. Asymptotics of a twoparameter generalization of Riemann’s zeta function 1.5. Fourier coeﬃcients of the product of two Hurwitz zeta functions 1.6. Several representations for the basic sum
1 3 3 5 6 7 7 9
Chapter 2. An Exact Representation for ζ(s)
11
Chapter 3. The Asymptotics of the Riemann Zeta Function for t ≤ η < ∞
15
Chapter 4. The Asymptotics of the Riemann Zeta Function for 0 < η < t
31
Chapter 5. Consequences of the Asymptotic Formulae
49
Part 2. Asymptotics to all Orders of a TwoParameter Generalization of the Riemann Zeta Function
53
Chapter 6. An Exact Representation for Φ(u, v, β)
55
Chapter 7. The Asymptotics of Φ(u, v, β)
59
Chapter 8. More Explicit Asymptotics of Φ(u, v, β)
73
Chapter 9. Fourier coeﬃcients of the product of two Hurwitz zeta functions 9.1. Corollaries 9.2. Asymptotics of the zeroth Fourier coeﬃcient 9.3. Proof of Theorem 9.1 9.4. Proof of Theorem 9.2 9.5. Fourier coeﬃcients of ζ(s, α) and ζ1 (s, α)
81 83 86 89 91 93
Part 3.
95
Representations for the Basic Sum
Chapter 10. Several Representations for the Basic Sum
97
Appendix A. The Asymptotics of Γ(1 − s) and χ(s)
107
Appendix B. Numerical Veriﬁcations B.1. Veriﬁcation of Theorem 3.1
109 109 v
vi
CONTENTS
B.2. B.3. B.4. B.5.
Veriﬁcation Veriﬁcation Veriﬁcation Veriﬁcation
Bibliography
of of of of
Theorem 3.2 Theorem 4.1 Corollary 4.3 Theorem 4.4
109 110 110 110 113
Abstract We present several formulae for the large t asymptotics of the Riemann zeta function ζ(s), s = σ + it, 0 ≤ σ ≤ 1, t > 0, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum ba n−s for certain ranges of a and b. In addition, we present precise estimates relating this sum with d the sum c ns−1 for certain ranges of a, b, c, d. We also study a twoparameter generalization of the Riemann zeta function which we denote by Φ(u, v, β), u ∈ C, v ∈ C, β ∈ R. Generalizing the methodology used in the study of ζ(s), we derive asymptotic formulae for Φ(u, v, β).
Received by the editor December 1, 2015, and, in revised form, July 6, 2018, October 26, 2018, and November 5, 2018. Article electronically published on December 21, 2021. DOI: https://doi.org/10.1090/memo/1351 2020 Mathematics Subject Classiﬁcation. Primary 11M06, 30E15; Secondary 33E20. Key words and phrases. Riemann zeta function, asymptotics. The authors are grateful to Professor D. Bump for several useful suggestions that led to a signiﬁcant improvement of the manuscript. They are also grateful to Kostis Kalimeris for many helpful remarks. The ﬁrst author is aﬃliated with the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, United Kingdom and was supported by the EPSRC, UK via a senior fellowship and by the Guggenheim Foundation, USA. He is deeply grateful to his students Michail Dimakos (partially supported by the Onassis Foundation) and Dionyssis Mantzavinos (partially supported by the EPSRC, UK) for performing a large number of numerical experiments during the early stages of this project. He also expresses his sincere gratitude to Anthony Ashton, Bryce McLeod, Joe Keller, David Stuart, and Eugene Shargorodsky, for extensive discussions and important suggestions. The second author is aﬃliated with the Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden and acknowledges support from the EPSRC, UK, the G¨ oran Gustafsson Foundation, Sweden, the Ruth and NilsErik Stenb¨ ack Foundation, Finland, the European Research Council, Grant Agreement No. 682537, and the Swedish Research Council, Grant No. 201505430. c 2022 American Mathematical Society
vii
Part 1
Asymptotics to all Orders of the Riemann Zeta Function
CHAPTER 1
Introduction It is well known, see for example theorem 4.11 in Titchmarsh [26], that the Riemann zeta function ζ(s), s = σ + it, σ, t ∈ R, satisﬁes the equation (1.1) ζ(s) =
1 1 x1−s − , + O ns 1−s xσ
t
1,
σ > 0,
t → ∞.
Here we present a systematic investigation of asymptotic formulae for ζ(s). In this connection, in chapter 2 we derive the following exact formula (see theorem 2.1): ζ(1 − s) =
η [ 2π ]
n
s−1
n=1
∞eiφ1 ∞eiφ2 s−1
iπs iπs 1 ηs z dz − + +e 2 − + e 2 , iπ iπ s − (2π) s ez − 1 ηe 2 ηe 2
(1.2) 0 < η < ∞,
−
π π < φj < , 2 2
j = 1, 2,
η η where [ 2π ] denotes the integer part of 2π and the contours of integration in the ﬁrst and second integrals in the rhs of equation (1.2) are the rays from η exp(−iπ/2) to ∞ exp(iφ1 ) and from η exp(iπ/2) to ∞ exp(iφ2 ), respectively.
1.1. The large t asymptotics of ζ(s) valid to all orders Equation (1.2) suggests a separate analysis for the cases t < η, t = η and t > η. The ﬁrst two cases are analysed in theorems 3.1 and 3.2 of chapter 3. Theorem 3.1 presents the large t asymptotics of ζ(s) valid to all orders in the case of (1 + )t < η. For example, the formulae of theorem 3.1 yield the following equation for the leading asymptotic terms in the case when (1 + )t < η for some > 0: η [ 2π ]
ζ(s) =
n−s −
n=1
(1.3)
+O
, 2+σ t2
η
1 η 1−s 2iη −s t − iσ iη iη Re Li + ) + (e ) −i arg(1 − e 2 1 − s 2π (2π)1−s η (1 + )t < η < ∞,
0 ≤ σ ≤ 1,
t → ∞,
where the error term is uniform for η, σ in the above ranges and the polylogarithm Lim (z) is deﬁned by Lim (z) =
∞ zk , km
k=1
3
m ≥ 1.
4
1. INTRODUCTION
Theorem 3.2 presents the large t asymptotics of ζ(s) valid to all orders in the case of η = t. The formula of theorem 3.2 yields the following equation for the leading asymptotic terms: 1−s t 1 1 − s 2π n=1
−it 2it−s eit iσ e [t + i(σ − 1)] − i Im Li1 (eit ) + Re Li2 (eit ) − − (2π)1−s 2t t 1 − Im Li3 (eit ) t
−s it 1 + i√ t e 1 + πt − i(3σ − 2) 1−s (2π) 2 3 i−1√ 2 1 + √ π 6σ − 6σ + 1 + 45σ 3 − 45σ 2 + 4 135t 24 t 1+i √ 4 3 2 − π 36σ − 24σ − 24σ + 12σ + 1 + O t−σ−2 , 3 576t 2 0 ≤ σ ≤ 1, t → ∞, t [ 2π ]
ζ(s) =
(1.4)
n−s −
where the error term is uniform for σ in the above range. The best estimate for the growth of ζ(s) as t → ∞, is not based on equation (1.1) but on the well known approximate functional equation, see for example equation (4.12.4) on page 79 of Titchmarsh [26]: 1 1 1 t −σ 2 −σ y σ−1 , xy = (1.5) , + χ(s) + O x + t ζ(s) = s 1−s n n 2π n≤x
n≤y
where 0 < σ < 1 and χ(s) is deﬁned by χ(s) =
πs (2π)s sin Γ(1 − s), π 2
with Γ(s) denoting the Gamma function. Building on Riemann’s unpublished notes, Siegel in his classical paper [24] presented the error termsof the rhs of equation (1.5) to all orders, only in the important case of x = y = t/2π. In theorems 4.1 and 4.4 we present analogous results for any x and y valid to all orders. Theorem 4.1 yields the following explicit √ result for the subleading term in the rhs of equation (1.5) when 1 < η = 2πy < t. For every > 0, there exists an A > 0 such that [t]
η
[ 2π ] iπ η 1 1 e−iπs Γ(1 − s) −([ ηt ]+1)iη iπ2 (s−1) s e 4 √ ζ(s) = e + χ(s) − e η ns n1−s 1 − e−iη i 2πt n=1 n=1 1 η , 1 < η < At 3 < ∞, O t5/6 −iπs − πt σ−1 +e Γ(1 − s)e 2 η × √ − At2 3 1 O e η + tη3/2 , t 3 < η < A t < ∞,
dist(η, 2πZ) > ,
0 ≤ σ ≤ 1,
t → ∞,
where the error term is uniform for all η, σ in√the above ranges. In corollary 4.3, similar formulas are presented for the case 2π t < η < 2π t.
1.2. THE EXPLICIT FORM OF CERTAIN SUMS
5
√ The case of t < η < t is analyzed in theorem 4.4, which provides a formula valid to all orders in terms of the function Φ(τ, u) deﬁned by (1.6)
2
Φ(τ, u) = 01
eπiτ x +2πiux dx, eπix − e−πix
τ < 0,
u ∈ C,
where 0 1 denotes a straight line parallel to e3πi/4 which crosses the real axis between 0 and 1. The formula of theorem 4.4 yields the following result for the leading asymptotic terms: For every > 0, [t]
η
[ 2π ] η 1 1 (1.7) ζ(s) = + χ(s) s n n1−s n=1 n=1
η 2 2t η it iπ(s−1) [ ]πi−it− 2η 2 (2[ 2π ]π−η) + e−iπs Γ(1 − s) e 2 η s−1 e η 2π σ πt η σ−1 η πi − iη Φ + O e− 2 5/6 × Φ+ , ∂2 Φ + 2 iη 2π t √
t < η < t, 0 ≤ σ ≤ 1, t → ∞,
where the error term is uniform for η, σ in the above ranges and Φ and ∂2 Φ are evaluated at the point 2πt 2t 2πt η t 1 − − − 2 , − 2 (1.8) . η η η 2π η 2 √ Remark 1.1. 1. In the particular case of η = 2πt, equation (1.7) agrees with the analogous formula of Siegel [24]. 2. Although Siegel suggested already in [24] the possibility of deriving a formula such as (1.7) based on the function (1.6), the authors have not been able to locate such a formula in the literature. In the case of σ = 1/2, an alternative asymptotic representation for ζ(s) involving a sum which is smoothly rather than sharply truncated is presented in [6]. An introduction to the RiemannSiegel formula can be found in chapter 7 of [8]. 2πt 3. Equation (1.7) is particularly useful in the case when η = b where b > 0 is a rational number. Indeed, in this case the function Φ and its derivatives evaluated at the point (1.8) can be computed explicitly (see equation (4.30) below). Therefore, in this case theorem 4.4 yields an explicit asymptotic expansion of ζ(s) to all orders. 4. Equation (1.3) is only useful as an asymptotic formula in the case when t/η → 0 as t → ∞ (otherwise the error term is as large as the retained terms). Therefore the asymptotic range where t < η and t/η = O(1) is not covered by this equation. However, this asymptotic sector is covered by corollary 4.3.
1.2. The explicit form of certain sums b Theorems 3.1 and 3.2 allow us to evaluate the sum a n−s , for certain a and b, to all orders, see theorems 5.1 and 5.2. For example, the leading order of such
6
1. INTRODUCTION
sums follows immediately from equations (1.3) and (1.4): For every > 0,
(1.9)
η2 [ 2π ]
n
−s
η
1 +1 n=[ 2π ]
η2 1−s η1 1−s 1 = − + O(η1−σ ), 1−s 2π 2π (1 + )t < η1 < η2 < ∞,
η 2π
(1.10)
[]
n
−s
t n=[ 2π ]+1
1 η 1−s 1 it πi = + e e4 1 − s 2π 2 (1 + )t < η < ∞,
0 ≤ σ ≤ 1, t 2π
12 −s
0 ≤ σ ≤ 1,
t → ∞,
+ O(t−σ ), t → ∞,
uniformly with respect to η1 , η2 , η, and σ.
1.3. The explicit form of the diﬀerence of certain sums It is well known, see for example page 78 of Titchmarsh [26], that for large t, the following sums coincide to the leading order:
n−s
and
χ(s) t 2πN
x 0,
and that the modiﬁed Hurwitz zeta function ζ1 (s, α) is deﬁned by ζ1 (s, α) = ζ(s, α) − α−s = ζ(s, α + 1). For each choice of u, v ∈ C \ {1}, the product ζ1 (u, α)ζ1 (v, α) is a smooth function of α ∈ [0, 1]; hence it can be represented by its Fourier series ζ1 (u, α)ζ1 (v, α) = (1.16) qn (u, v)e2πinα , n∈Z
where the Fourier coeﬃcients qn (u, v) are deﬁned by 1 qn (u, v) = ζ1 (u, α)ζ1 (v, α)e−2πinα dα,
n ∈ Z.
0
For u = v¯ = σ + it, the zeroth Fourier coeﬃcient q0 (u, v) is given by the mean square average of the modiﬁed Hurwitz function: 1 q0 (σ + it, σ − it) = ζ1 (σ + it, α)2 dα. 0
Attempts to compute the large t asymptotics of q0 (σ +it, σ −it) have a long history, especially in the case when σ = 1/2. Increasingly reﬁned asymptotic estimates were derived in an extensive series of papers (see [4, 14, 17, 21, 27, 28]) before the following formula was ﬁnally obtained independently in [1] and [29]: 2 1 1 ζ1 1 + it, α dα = ln t + γ − 2 Re ζ( 2 + it) + O(t−1 ), (1.17) 1 2 2π 0 2 + it where γ denotes the Euler constant. An alternative derivation of (1.17) was presented in [15]. In the derivation of [15], the asymptotic formula (1.17) is obtained as an easy consequence of the following identity [15, Eq. (2.1) with N = 1]: (1.18)
q0 (u, v) =
1 + R0 (u, v) + R0 (v, u) − T0 (u, v) − T0 (v, u), u+v−1
where 0 < Re u, Re v < 2 and R0 (u, v) and T0 (u, v) are deﬁned by Γ(1 − v) , Γ(u) ∞ ∞ l1−u−v β u+v−2 (1 + β)−u−1 dβ.
(1.19)
R0 (u, v) = Γ(u + v − 1)ζ(u + v − 1)
(1.20)
T0 (u, v) =
u ζ(u) − 1 + 1−v 1−v
l=1
l
Actually, a version of the identity (1.18) is presented as the main result of [15], because from this identity many asymptotic results (such as (1.17)) can easily be obtained. A key point here is that, for u = σ + it and v = σ − it, the sum appearing in the deﬁnition (1.20) of T0 (u, v) is small as t → ∞. In chapter 9, we will analyze the large t asymptotics of the nth Fourier coeﬃcient qn for any n ∈ Z by establishing a generalization of (1.18). We will ﬁrst show
1.6. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
that R0 and T0 are the analytic continuations of the integrals ∞ (1.21a) α−v ζ1 (u, α)dα, Re(u + v) > 2, R0 (u, v) = 0 1
(1.21b)
T0 (u, v) =
α−v ζ1 (u, α)dα,
9
Re v < 1,
Re v < 1.
0
Then, we will prove (see Theorem 9.1) the following generalization of (1.18) which is valid for n ∈ Z and Re u, Re v < 2: (1.22)
qn (u, v) = bn (u + v) + Rn (u, v) + Rn (v, u) − Tn (u, v) − Tn (v, u),
where, for any n ∈ Z, bn (s), Rn (u, v), and Tn (u, v) are the analytic continuations of the following functions: ∞ bn (s) = α−s e−2πinα dα, Re s > 1, 1 ∞ Rn (u, v) = α−v ζ1 (u, α)e−2iπnα dα, Re(u + v) > 2, Re v < 1,
0 1
Tn (u, v) =
α−v ζ1 (u, α)e−2πinα dα,
Re v < 1.
0 1 Since b0 (s) = s−1 , it follows immediately from (1.21) that equation (1.22) reduces to (1.18) when n = 0. Just like the identity (1.18) can be used to ﬁnd the large t asymptotics of the zeroth Fourier coeﬃcient q0 , the identity (1.22) can be used to ﬁnd the large t behavior of the nth Fourier coeﬃcient qn . The function Φ(u, v, β) enters in this formulation, because it turns out (see equation (9.6b)) that for n = 0 the function Rn (u, v) is given by the following analog of (1.19): πiv ie n ≥ 1, Γ(1 − v) sin(πu) , Φ(u, v, n) × 2ie Rn (u, v) = −πiv Γ(u) n ≤ −1. 2 sin(πu) ,
We will show in chapter 9 that the asymptotics of Tn can be obtained (as in the case n = 0) for any n = 0. Thus, the computation of the leading asymptotics of the Fourier coeﬃcients qn , n = 0, reduces to the calculation of the large t asymptotics of Φ(u, v, n) for integer values of n. By substituting the asymptotic results for Φ obtained in chapter 7 into (1.22), we obtain in chapter 9 an asymptotic formula for the nth Fourier coeﬃcient of ζ1 (σ + it, α)2 . These considerations also lead to Lindel¨of type asymptotic bounds on the function Φ(u, v, n) for large t (see Corollary 9.9): ⎧ ⎪ σ ∈ [0, 1/2), ⎨O(1), n ≥ 1, t → ∞. Φ(σ + it, σ − it, n) = O(ln t), σ = 1/2, ⎪ ⎩ 2σ−1 ), σ ∈ (1/2, 1], O(t 1.6. Several representations for the basic sum The remarkable fact about the large tasymptotics of the Riemann zeta function is that, whereas the higher order terms in the asymptotic expansion can be computed explicitly, the sum appearing in the leading order is transcendental. In chapter 10 we present several integral representations of this basic sum with the
10
1. INTRODUCTION
hope that these representations may be useful for the estimation of this fundamental sum (some results in this direction can be found in [2]). Let Cηt denote the semicircle from iη to it with Re z ≥ 0:
i(η + t) t − η iθ π π + e − δ, dist(t, 2πZ) > δ, and the contour in the second equality above denotes the principal value integral with respect to the points
t η +1≤n≤ . 2πn n ∈ Z, 2π 2π Basic Assumption: In the statements of all results throughout this work, it will be assumed (also when not explicitly mentioned) that (1.24)
η > 0,
t > 0,
η∈ / 2πZ,
t∈ / 2πZ.
CHAPTER 2
An Exact Representation for ζ(s) In chapters 2 and 3, we assume that the branch cut for the logarithm runs along the negative real axis. By the standing assumption (1.24), we always have η, t ∈ / 2πZ. Theorem 2.1 (An exact representation for Riemann’s zeta function). Let ζ(s), s = σ + it, σ, t ∈ R, denote the Riemann zeta function. Then, η [ 2π ] ns−1 ζ(s) = χ(s) n=1
(2.1)
∞eiφ1 ∞eiφ2 −z s−1 iπs 1 ηs e z dz − iπs 2 2 + +e − + e , iπ iπ s − (2π) s 1 − e−z ηe 2 ηe 2 π π 0 < η < ∞, − < φj < , j = 1, 2, 2 2
where
πs (2π)s sin Γ(1 − s), π 2 with Γ(s), s ∈ C, denoting the Gamma function, and where the contours of the ﬁrst and second integrals in the rhs of ( 2.1) are the rays from η exp(−iπ/2) to ∞ exp(iφ1 ) and from η exp(iπ/2) to ∞ exp(iφ2 ) respectively. An equivalent formula is η [ ∞eiφ1 ∞eiφ2 −z s−1
2π ] iπs e z dz 1 ηs s−1 − iπs 2 2 n + +e − + e , ζ(1 − s) = iπ iπ s − (2π) s 1 − e−z ηe 2 ηe 2 n=1 (2.2)
χ(s) =
(2.3) 0 < η < ∞, −
π π < φj < , j = 1, 2. 2 2
Proof. We will ﬁrst derive equation (2.1) in the particular case 0 < η < 2π. In this case, replacing η with α, (2.1) can be written as ∞eiφ1 ∞eiφ2 −z s−1 iπs iπs e z χ(s) αs − ζ(s) = +e 2 dz , − + e 2 iπ iπ (2π)s s 1 − e−z αe− 2 αe 2 π −π < φj < , j = 1, 2. 2 2 We decompose the Hankel contour Hα in the expression (1.14) for ζ(s) into the union of three diﬀerent contours (see ﬁgure 2.1), namely
(2.4)
0 < α < 2π;
Hα = L 1 ∪ L 2 ∪ L 3 , 11
12
2. AN EXACT REPRESENTATION FOR ζ(s)
iα
L1
L3 Re z
L2
−iα
Figure 2.1. The decomposition of Hα into L1 + L2 + L3 . iα 1 3 2 −iα Figure 2.2. The domains 1, 2, and 3.
where
π ! L1 = αeiθ < θ < π ∪ {zeiπ  α < z < ∞)}, 2 π! −iπ L2 = {ze  α < z < ∞} ∪ αeiθ − π < θ < − , 2 ! π π (2.5) . L3 = αeiθ − < θ < 2 2 The integral along L1 can be written as follows: ∞eiπ ∞ z s−1 ez e−u us−1 du s−1 iπs dz = (2.6) z dz = e , iπ iπ −z − 1 1 − ez 1 − e−u L1 e αe 2 αe− 2 where the contours of the second and third integrals in (2.6) are the rays in the cut complex zplane from α exp(iπ/2) to ∞ exp(iπ) and from α exp(−iπ/2) to ∞ respectively. Indeed, in the domain enclosed by L1 and by the ray from α exp(iπ/2) to ∞ exp(iπ), i.e., in the shaded domain 1 of ﬁgure 2.2, we have π z = reiθ , α ≤ r ≤ ∞, ≤ θ ≤ π; 2 thus, ez  = er cos θ , α ≤ r ≤ ∞, 0 ≤ cos θ ≤ −1. Hence, Cauchy’s theorem implies the ﬁrst equality of (2.6). Then, the substitution u = z exp(−iπ), yields the second equality in (2.4). Similarly, (2.7) L2
z s−1 dz = e−z − 1
αe−
iπ 2
∞e−iπ
ez z s−1 dz = e−iπs 1 − ez
αe
∞
iπ 2
e−u us−1 du, 1 − e−u
2. AN EXACT REPRESENTATION FOR ζ(s)
13
where the curve L2 is deﬁned in (2.5) and the contours of the second and third integrals in (2.7) are the rays in the cut complex zplane from ∞ exp(−iπ) to α exp(−iπ/2) and from ∞ to α exp(iπ/2) respectively. Indeed, the ﬁrst equality in (2.7) is the consequence of Cauchy’s theorem applied in the domain enclosed by L2 and by the ray from ∞ exp(−iπ) to α exp(−iπ/2) (i.e., in the shaded domain 2 of ﬁgure 2.2), whereas the second equality in (2.7) follows from the substitution u = z exp(iπ). The integral along L3 can be written as follows: ⎛ ⎞ ∞ αe iπ2 πs αs s−1 −z z ⎠ e (2.8) dz = − ⎝ , + z s−1 dz − 2i sin iπ −z −z −1 1−e 2 s L3 e ∞ αe− 2 where the curve L3 is deﬁned in (2.5) and the contours in the ﬁrst and second integrals in (2.8) are the rays in the cut complex zplane from α exp(−iπ/2) to ∞ and from ∞ to α exp(iπ/2). Indeed, z s−1 e−z s−1 dz = − (2.9) z dz − z s−1 dz. −z − 1 −z e 1 − e L3 L3 L3 The second term in the rhs of (2.9) yields the second term in the rhs of (2.8). Furthermore, in the domain enclosed by L3 and by the two rays (α exp(−iπ/2), ∞) and (∞, α exp(iπ/2)), i.e., in the shaded domain 3 of ﬁgure 2.2, we have π π z = reiθ , α ≤ r ≤ ∞, − < θ < ; 2 2 thus e−z  = e−r cos θ , α ≤ r ≤ ∞, cos θ ≥ 0. Hence, Cauchy’s theorem implies that the ﬁrst term of the rhs of (2.9) equals the ﬁrst term in the rhs of (2.8). Adding equations (2.6)–(2.8) we ﬁnd equation (2.4) but with φ1 = φ2 = 0. However, Cauchy’s theorem implies that the rays (α exp(−iπ/2), ∞) and (∞, α exp(iπ/2)) can be replaced with the rays (α exp(−iπ/2), ∞ exp(iφ1 )) and
(∞ exp(iφ2 ), α exp(iπ/2)),
respectively, and then we ﬁnd (2.4). In order to derive equation (2.1) with 2π < η < ∞, we introduce the curves −α Cαη , C−η and Cˆηα , which are the following semicircles in the complex zplane, see ﬁgure 2.3:
i(η + α) (η − α) iθ π π + e − < θ < Cαη = (2.10a) , 2 2 2 2
−i(η + α) (η − α) iθ π π −α + e − < θ < = (2.10b) , C−η 2 2 2 2
i(η + α) (η − α) iθ π π + e −π < θ < ∪ 0, 0 ≤ σ ≤ 1, N ≥ 2,
t → ∞,
where the error term is uniform for all η, , σ, N in the above ranges and (2N + 1)!! is deﬁned by (2N + 1)!! = 1 · 3 · 5 · · · · · (2N − 1)(2N + 1). For N = 3 equation ( 3.1) simpliﬁes to η [ 2π ]
(3.2)
ζ(s) =
1 η 1−s 1 − s 2π n=1
−s 2iη t − iσ + −i arg(1 − eiη ) + Re Li2 (eiη ) 1−s (2π) η ' 1& + 2 it2 + (2σ + 1)t − iσ(σ + 1) Im Li3 (eiη ) η 1 + 8 1 + O 3+σ t3 + , η
(1 + )t < η < ∞, > 0, 0 ≤ σ ≤ 1, t → ∞,
n−s −
15
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
16
where the error term is uniform for all η, , σ in the above ranges and the polylogarithm Lim (z) is deﬁned by Lim (z) =
(3.3)
∞ zk , km
m ≥ 1.
k=1
Similarly, it is straightforward for any N ≥ 4 to derive an asymptotic formula for ζ(s) analogous to ( 3.2) with an error term of order 1 + 2(N +1) 1 N O N +σ t + . η
Proof. Suppose ﬁrst that 0 < t ≤ η < ∞, > 0, 0 ≤ σ ≤ 1, N ≥ 2. All error terms of the form O(·) will be uniform with respect to η, , σ, N unless otherwise speciﬁed. The proof is based on equation (2.3), i.e., η [ 2π ]
(3.4) ζ(1 − s) =
ns−1 −
n=1
1 η s + GL (t, σ; η) + GU (t, σ; η), s 2π
where GL and GU are deﬁned by iπs ∞eiφ1 e−z z s−1 dz e 2 (3.5) GL (t, σ; η) = , (2π)s −iη 1 − e−z
0 < η < ∞,
−
π π < φ1 < , 2 2
0 < η < ∞,
−
π π < φ2 < , 2 2
0 < η < ∞.
and e− 2 (3.6) GU (t, σ; η) = (2π)s iπs
∞eiφ2 iη
e−z z s−1 dz , 1 − e−z
The assumption η ∈ / 2πZ implies that GL and GU are welldeﬁned. The asymptotics of GL Using the expansion (3.7)
∞ 1 = e−nz , 1 − e−z n=0
Re z > 0,
in the deﬁnition (3.5) of GL , we ﬁnd (3.8)
iφ1 iπs ∞ e 2 ∞e e−nz z s−1 dz, GL (t, σ; η) = (2π)s n=1 −iη
−
π π < φ1 < , 2 2
0 < η < ∞.
The interchange of the integration and the summation in (3.8) is allowed because the sum on the rhs of (3.7) converges absolutely and uniformly in Re z ≥ δ for any δ > 0. Integration by parts shows that the integral in (3.8) can be written as j ∞eiφ1 N −1 σ−1 1 d z e−nz z s−1 dz = e−nz+it ln z it dz it n− z n− z −iη j=0 z=−iη ∞eiφ1 (3.9) + e−nz+it ln z DN dz, −iη
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
17
where DN = DN (z, n, σ, t) is shorthand notation for N 1 d DN := (3.10) z σ−1 . dz n − it z We claim that
iφ1 iπs ∞ e 2 ∞e −nz+it ln z 2 σ−1−N e D dz = O (2N − 1)!!(N + 1) η . N (2π)s n=1 −iη
(3.11)
In order to derive equation (3.11) we ﬁrst note that we can write DN in the form DN =
(3.12)
N N b=0 c=0
where
(N ) Abc ,
(N )
Abc z σ−1
tb (nz)N −b σ c (nz − it)2N
b = 0, . . . , N , c = 0, . . . , N , are integers which satisfy
(N )
Abc  ≤ (2N − 1)!! = 1 · 3 · 5 · · · · · (2N − 1),
b = 0, . . . , N,
c = 0, . . . , N.
Indeed, if DN can be written as in (3.12), then the computation b N −b c 1 σ d (N ) σ−1 t (nz) Abc z dz n − it (nz − it)2N z (N )
= Abc z σ−1
tb (nz)N −b σ c [it(b − N ) − itσ − nz(1 + b + N ) + nzσ)] (nz − it)2(N +1)
shows that DN +1 also can be written as in (3.12). In order to estimate the lhs of (3.11), we choose φ1 = 0. Then, since t ≤ z and n − it z  ≥ n for all z on the contour, we can estimate N N t b (nz)N σ c DN = O (2N − 1)!!zσ−1 nz 2N z 2N (n − it z) b=0 c=0 1 = O (2N − 1)!!zσ−1−N (N + 1)2 N , z = u − iη, u ∈ [0, ∞). n Thus, the lhs of (3.11) is ∞ ∞ − πt −nz it ln z 2 σ−1−N 1 2 e e (2N − 1)!!(N + 1) z dz O e nN n=1 −iη ∞ ∞ 1 2 −n Re z σ−1−N = O (2N − 1)!!(N + 1) e z dz nN −iη n=1 ∞ ∞ 1 −nu = O (2N − 1)!!(N + 1)2 η σ−1−N e du nN 0 n=1 ∞ 1 = O (2N − 1)!!(N + 1)2 η σ−1−N nN +1 n=1 = O (2N − 1)!!(N + 1)2 η σ−1−N . This proves (3.11).
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
18
Equations (3.8), (3.9), and (3.11) imply that GL satisﬁes (3.13)
j iπs ∞ N −1 d z σ−1 e 2 −nz+it ln z 1 e GL (t, σ; η) = (2π)s n=1 j=0 dz n − it n − it z z z=−iη + O (2N − 1)!!(N + 1)2 η σ−1−N , t ≤ η < ∞, t → ∞.
The asymptotics of GU We now let > 0 and suppose that 0 < (1 + )t < η < ∞. In analogy with equations (3.8) and (3.9) we now ﬁnd
(3.14)
iφ2 iπs ∞ e− 2 ∞e GU (t, σ; η) = e−nz z s−1 dz, (2π)s n=1 iη
−
π π < φ2 < , 2 2
0 < η < ∞,
and
∞eiφ2
−nz s−1
e iη
z
dz =
N −1
e
j=0
(3.15)
−nz+it ln z
∞eiφ2
+
1 n−
it z
d dz
j
z σ−1 n − it z z=iη
e−nz+it ln z DN dz,
iη
where DN is deﬁned by (3.10). We claim that
(3.16)
iφ2 iπs ∞ 1 + 2N e− 2 ∞e −nz+it ln z σ−N e DN dz = O (2N − 1)!!N η . (2π)s n=1 iη
Indeed, since DN can be written in the form given by (3.12), Jordan’s lemma implies that we may choose φ2 = π/2. Then, using the estimate N N t b (nz)N σ c DN = O (2N − 1)!!zσ−1 nz 2N z 2N (n − it z) b=0 c=0 nN = O (2N − 1)!!zσ−1−N (N + 1)2 , z = iλ, (n − ηt )2N
λ ∈ [η, ∞),
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
19
we ﬁnd that the lhs of (3.16) is ∞ πt O e2 n=1
i∞
e−nz+it ln z DN dz
iη
∞ πt =O e2 n=1
i∞
e−
πt 2
nN dz (n − ηt )2N N
(2N − 1)!!zσ−1−N (N + 1)2
iη
2 = O (2N − 1)!!(N + 1)
∞
∞
n 1 2N (n − η 1+ ) n=1 σ−N 1 + 2N 2 η = O (2N − 1)!!(N + 1) N −σ
1 + 2N = O (2N − 1)!!N η σ−N .
λ
σ−1−N
dλ
In the third equality above we have used the fact that
(3.17)
∞ n=1
nN (n −
1 2N 1+ )
= ≤
∞ 1 nN (1 − n=1
1 (1 −
1 2N 1+ )
1 1 2N (1+)n ) ∞
1 + 2N 1 = O , nN
n=1
N ≥ 2.
This proves (3.16). We next claim that (3.16) can be modiﬁed as follows:
(3.18)
iφ2 iπs ∞ e− 2 ∞e e−nz+it ln z DN dz (2π)s n=1 iη 1 + 2(N +1) σ−N −1 η . = O (2N + 1)!!N
Indeed, integration by parts shows that the lhs of (3.16) equals iπs ∞ e− 2 −nz+it ln z 1 e (2π)s n=1 n−
D N it z
z=iη
+
iφ2 iπs ∞ e− 2 ∞e e−nz+it ln z DN +1 dz. (2π)s n=1 iη
By (3.16), the second term is 1 + 2(N +1) σ−N −1 η , O (2N + 1)!!(N + 1)
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
20
while, in view of (3.12) and (3.17), the ﬁrst term satisﬁes iπs ∞ N N e− 2 −niη+it ln iη 1 (N ) tb (niη)N −b σ c e Abc (iη)σ−1 t s (2π) n=1 (niη − it)2N n − η b=0 c=0 N ∞ N tb nN −b (N ) = O η σ−1−N Abc  η b (n − ηt )2N +1 n=1 b=0 c=0 ∞ nN σ−1−N 2 =O η (2N − 1)!!(N + 1) 1 2N +1 (n − 1+ ) n=1 1 + 2N +1 = O η σ−1−N (2N − 1)!!(N + 1)2 .
This proves (3.18). Equations (3.14), (3.15), and (3.18) imply that GU satisﬁes (3.19)
j iπs ∞ N −1 1 d z σ−1 e− 2 −nz+it ln z GU (t, σ; η) = e (2π)s n=1 j=0 dz n − it n − it z z z=iη 1 + 2(N +1) η σ−N −1 , + O (2N + 1)!!N (1 + )t < η < ∞.
Proof of (3.1) Substituting the expressions (3.13) and (3.19) for GL and GU into (3.4), we ﬁnd η [ 2π ]
ζ(1 − s) =
n=1
ns−1 −
1 η s s 2π
iπs ∞ N −1 e 2 −nz+it ln z + e (2π)s n=1 j=0
1 n−
it z
d dz
j
z σ−1 n − it z z=−iη
j iπs ∞ N −1 d z σ−1 1 e− 2 −nz+it ln z + e (2π)s n=1 j=0 dz n − it n − it z z z=iη 1 + 2(N +1) + O (2N + 1)!!N η σ−N −1 ,
(1 + )t < η < ∞, t → ∞. Replacing σ by 1 − σ and taking the complex conjugate of the resulting equation, we ﬁnd (3.1).
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
21
Proof of (3.2) Letting N = 3 in (3.1), we ﬁnd η [ 2π ]
ζ(s) =
n=1
n−s −
1 η 1−s 1 − s 2π
iπ(1−s) ∞ 1 1 i(q(σ − 1) + σ) e− 2 −s −iηn 1 + 2 + (iη) e 1−s (2π) n 1 + q ηn (1 + q)3 n=1 1 −q 2 (σ − 1)2 + q −2σ 2 + σ + 2 − σ(σ + 1) + 2 3 η n (1 + q)5
iπ(1−s) ∞ 1 1 1 i(q(σ − 1) − σ) e 2 −s + 2 + (−iη) eiηn (2π)1−s n 1 − q ηn (1 − q)3 n=1 1 q 2 (σ − 1)2 + q −2σ 2 + σ + 2 + σ(σ + 1) − 2 3 η n (1 − q)5 1 + 8 t + O η −σ−3 . , q=
nη The above sums can be expanded as series in q with coeﬃcients expressible in terms of the polylogarithms Lim (z) deﬁned by (3.3). For example, k ∞ ∞ eiηn eiηn t = n(1 − q) n=1 n nη n=1 k=0 ∞ ∞ k t eiηn = η nk+1 n=1 k=0 ∞ k t = Lik+1 eiη , η ∞
k=0
where the interchange of the two sums can be justiﬁed by separating the terms where k = 0 and noticing that the remaining double sum is absolutely convergent. More precisely, N k k M M N N eiηn eiηn t eiηn t + lim = lim lim lim N →∞ M →∞ N →∞ M →∞ n nη n n nη n=1 n=1 n=1 k=0 k=1 k ∞ ∞ eiηn t = Li1 (eiη ) + n nη n=1 k=1 ∞ ∞ t k eiηn = Li1 (eiη ) + η nk+1 k=1 n=1 ∞ k t = Lik+1 eiη . η k=0
22
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
Similarly, for any j ≥ 0 and l, m ≥ 1,
k+j ∞ ∞ ∞ t eiηn qj eiηn k + m − 1 = l (1 − q)m l k n n nη n=1 n=1 k=0 k+j ∞ k+m−1 t = Lik+l+j (eiη ). k η k=0
Letting η → −η in this equation, we ﬁnd k+j ∞ ∞ k+m−1 e−iηn (−q)j t = Lik+l+j (e−iη ). − l (1 + q)m k n η n=1 k=0
This gives the following expression for the Riemann zeta function: η [ 2π ]
1 η 1−s 1 − s 2π n=1
k k iπ(1−s) ∞ ∞ e− 2 iσ k + 2 t t −s −iη + (iη) Li (e ) + Lik+2 (e−iη ) − − k+1 k (2π)1−s η η η k=0 k=0 k+1 ∞ k+2 i(σ − 1) t Lik+3 (e−iη ) − − k η η k=0 k ∞ σ(σ + 1) k + 4 t Lik+3 (e−iη ) − − k η2 η k=0 k+1 ∞ 2 2σ − σ − 2 k + 4 t Lik+4 (e−iη ) + − k η2 η k=0 k+2 ∞ (σ − 1)2 k + 4 t −iη Li (e ) − − k+5 k η2 η k=0
k iπ(1−s) ∞ ∞ k e 2 t t iσ k + 2 −s iη + (−iη) Lik+1 (e ) − Lik+2 (eiη ) k (2π)1−s η η η k=0 k=0 k+1 ∞ t i(σ − 1) k + 2 Lik+3 (eiη ) + k η η k=0 k ∞ t σ(σ + 1) k + 4 − Lik+3 (eiη ) k η2 η k=0 k+1 ∞ t 2σ 2 − σ − 1 k + 4 Lik+4 (eiη ) + 2 k η η k=0 k+2 ∞ 2 k+4 t (σ − 1) iη Lik+5 (e ) − k η2 η k=0 1 + 8 + O η −σ−3 .
ζ(s) =
n−s −
By including only the ﬁrst few terms in the sums over k, we ﬁnd an expansion for ζ(s) in powers of ηt . For example, including only the terms of order larger than
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
23
3
t O( η3+σ ) we have
η [ 2π ]
ζ(s) =
1 η 1−s 1 − s 2π n=1 2
t iη −s t −iη −iη − Li3 (e−iη ) Li1 (e ) − Li2 (e ) + 1−s (2π) η η iσ 3iσt + Li2 (e−iη ) − 2 Li3 (e−iη ) η η i(σ − 1)t σ(σ + 1) −iη −iη + Li (e ) − Li (e ) 3 3 η2 η2
2 t iη −s t iη iη + Li3 (eiη ) Li1 (e ) + Li2 (e ) + (2π)1−s η η iσ 3iσt − Li2 (eiη ) − 2 Li3 (eiη ) η η i(σ − 1)t σ(σ + 1) iη iη + Li (e ) − Li (e ) 3 3 η2 η2 1 + 8 1 + O 3+σ t3 + . η
n−s −
After simpliﬁcation we ﬁnd
η [ 2π ]
ζ(s) =
1 η 1−s 1 − s 2π n=1 2
t 2iη −s t iη iη Re Li + (e ) + (e ) + i Im Li3 (eiη ) i Im Li 1 2 (2π)1−s η η iσ 3σt − Re Li2 (eiη ) + 2 Im Li3 (eiη ) η η (σ − 1)t iσ(σ + 1) iη iη − Im Li3 (e ) − Im Li3 (e ) η2 η2 1 + 8 1 + O 3+σ t3 + , η
n−s −
The identity Li1 eiη = − ln(1 − eiη ) now yields (3.2).
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
24
Theorem 3.2 (The asymptotic expansion to all orders for the case t = η). Let ζ(s), s = σ + it, σ, t ∈ R, denote the Riemann zeta function. Then, t [ 2π ]
ζ(s) =
n=1
n−s −
1 1−s
t 2π
1−s
j iπ(1−s) ∞ N −1 1 d z −σ e− 2 −nz−it ln z + e (2π)1−s n=1 j=0 dz n + it n + it z z z=it j iπ(1−s) ∞ N −1 d z −σ e 2 1 −nz−it ln z + e (2π)1−s n=2 j=0 dz n + it n + it z z z=−it 1−s 2N ck (1 − σ)Γ( k+1 t it 2N −σ−N 2 ) + e + O (2N + 1)!!N 2 t , k+1 2π t 2 k=0 0 ≤ σ ≤ 1,
(3.20)
N ≥ 2,
t → ∞,
where the error term is uniform for all σ, N in the above ranges and the coeﬃcients ck (σ) are given by equation ( 3.30) below. The ﬁrst few of the ck ’s are given by 1−i , 2 i c1 (σ) = − iσ, 3 1+i 2 c2 (σ) = − 6σ − 6σ + 1 , 12 (3.21) 1 −45σ 3 + 90σ 2 − 45σ + 4 , c3 (σ) = 135 i−1 36σ 4 − 120σ 3 + 120σ 2 − 36σ + 1 , c4 (σ) = 432 i c5 (σ) = 189σ 5 − 945σ 4 + 1575σ 3 − 987σ 2 + 168σ + 8 , 5670 1+i c6 (σ) = 1080σ 6 − 7560σ 5 + 18900σ 4 − 20160σ 3 + 8190σ 2 − 450σ − 139 . 194400 c0 (σ) =
For N = 3 equation ( 3.20) simpliﬁes to 1−s t 1 ζ(s) = n − 1 − s 2π n=1
−it −s it ' e & 2 2it e iσ Re Li2 (eit ) −t − it(σ − 1) + (σ − 1)2 − i Im Li1 (eit ) + − 1−s 2 (2π) 2t t it + σ + σ 2 2 + 3σ 3i it it it + i Im Li3 (e ) + Re Li4 (e ) + 2 Im Li5 (e ) t2 t2 t t [ 2π ]
−s
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
(3.22)
t−s eit 1 + i √ 1 + πt − i(3σ − 2) (2π)1−s 2 3 √ i−1 1 + √ π 6σ 2 − 6σ + 1 + 45σ 3 − 45σ 2 + 4 135t 24 t 1+i √ π 36σ 4 − 24σ 3 − 24σ 2 + 12σ + 1 − 3 576t 2 i 189σ 5 − 315σ 3 + 42σ 2 + 84σ − 8 + 2 2835t 1−i √ π 1080σ 6 + 1080σ 5 − 2700σ 4 − 1440σ 3 + 5 103680t 2 2 0 ≤ σ ≤ 1, + 1710σ + 270σ − 139 + O t−σ−3 ,
25
t → ∞,
where the error term is uniform for all σ in the above range and the polylogarithm Lim (z), m ≥ 1, is deﬁned by ( 3.3). Similarly, it is straightforward for any N ≥ 4 to derive an asymptotic formula for ζ(s) analogous to ( 3.22) with an error term of order O t−σ−N . Proof. Setting η = t in (3.4), we ﬁnd t [ 2π ]
(3.23) ζ(1 − s) =
n
s−1
n=1
1 − s
t 2π
s + GL (t, σ; t) + GU (t, σ; t),
0 < t < ∞.
Equation (3.13) is valid also when η = t and gives the asymptotics of GL (t, σ; t). On the other hand, using the expansion (3.7), we write (1)
(2)
GU (t, σ; t) = GU (t, σ; t) + GU (t, σ; t), where e− 2 = (2π)s iπs
(3.24)
(1) GU (t, σ; t)
∞eiφ2
e−z z s−1 dz,
π π < φ2 < , 2 2
0 < t < ∞,
π π < φ2 < , 2 2
0 < t < ∞.
−
it
and (3.25) (2)
GU (t, σ; t) =
iφ2 iπs ∞ e− 2 ∞e e−nz z s−1 dz, (2π)s n=2 it
(2)
−
The asymptotics of GU can be found using integration by parts, whereas the (1) asymptotics of GU will be computed by considering the critical point at z = it. (2) The asymptotics of GU Repeating the steps that led to (3.19) but with the sum over n only going from 2 to ∞ and with t = η, we ﬁnd the following analog of (3.19): j iπs ∞ N −1 d z σ−1 e− 2 −nz+it ln z 1 (2) e GU (t, σ; t) = (2π)s n=2 j=0 dz n − it n − it z z z=it + O (2N + 1)!!N 22N tσ−N −1 . (3.26)
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
26
(1)
The asymptotics of GU (1) Letting in the deﬁnition (3.24) of GU , φ2 = 0 and ρ ∈ [0, ∞),
z = it + tρ, we ﬁnd e− 2 = (it)s−1 te−it (2π)s iπs
(1) GU
∞
e−t(ρ−i ln(1−iρ)) (1 − iρ)σ−1 dρ.
0
Letting v = ρ − i ln(1 − iρ),
(3.27) we ﬁnd
(1)
GU =
t 2π
s
e−it
γ
e−tv
(1 − iρ(v))σ dv, ρ(v)
where ρ(v) is deﬁned by inverting (3.27) and γ denotes the image of the contour [0, ∞) under (3.27). In order to ascertain that the value of ρ(v) is welldeﬁned, we consider in detail the map φ deﬁned by φ : C \ [−i, −i∞) → C, ρ → v = ρ − i ln(1 − iρ), where [−i, −i∞) is a branch cut and the principal branch is chosen for the logarithm, i.e., (3.28)
v = ρ − i ln 1 − iρ + arg(1 − iρ),
arg(1 − iρ) ∈ (−π, π).
The function φ satisﬁes φ(ρ) = 0 iﬀ ρ = 0; ρ φ (ρ) = = 0 iﬀ ρ = 0; i+ρ iρ2 + O(ρ3 ), v = φ(ρ) = − ρ → 0. 2 Moreover, we claim that φ maps the ﬁrst quadrant of the complex ρplane bijectively onto the region delimited by the contour γ = φ([0, ∞)) and the positive imaginary axis in the complex vplane, see ﬁgure 3.1. Indeed, it is clear that φ maps [0, i∞) bijectively onto [0, i∞). Writing ρ = ρ1 + iρ2 , we have ∂ Re φ ρ2 + ρ2 ∂ Re φ ρ1 = 2 , = 2 , ∂ρ1 ρ1 + (ρ2 + 1)2 ∂ρ2 ρ1 + (ρ2 + 1)2 This shows that each level curve of the function Re φ in the ﬁrst quadrant intersects the positive ρaxis in a unique point and that the vector −ρ1 , ρ2 + ρ2 (3.29) is tangent to the level curve passing through ρ. Since ∂ Im φ −ρ1 = 2 , ∂ρ1 ρ1 + (ρ2 + 1)2
∂ Im φ ρ2 + ρ2 = 2 , ∂ρ2 ρ1 + (ρ2 + 1)2
we infer that the derivative of Im φ in the direction of (3.29) equals ρ2 , showing that Im φ is strictly increasing along each level curve. This proves the claim. Thus, if v belongs to the region delimited by the contour γ = φ([0, ∞)) and the positive imaginary axis, we may deﬁne ρ(v) as the unique inverse image of v under
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
Im ρ
27
Im v
φ
Re ρ −i
Re v −i
γ
Figure 3.1. The function φ maps the indicated shaded areas in the complex ρplane bijectively onto the corresponding shaded areas in the vplane. φ which belongs to the ﬁrst quadrant. Using analyticity to deform the contour γ to the positive real axis, we obtain s ∞ t (1 − iρ(v))σ (1) −it dv. e e−tv GU = 2π ρ(v) 0 −1 We claim that there exist coeﬃcients {ck (σ)}N such that 0 N −1 √ (1 − iρ(v))σ 1 =√ (3.30) ck (σ)v k/2 + rN ( v) , v ≥ 0, ρ(v) v k=0 √ where rN ( v) satisﬁes N +1 √ ∞ Γ( 2 ) −tv rN ( v) √ dv = O e (3.31) . N +1 v t 2 0 Indeed, since φ has a double zero at ρ = 0, there exists a neighborhood V of ρ = 0 in the complex ρplane and a Riemann surface Σ which is a√twosheeted cover √ of V with local parameter λ := v, such that the map ρ → v is bijective and holomorphic V → Σ. The function φ is analytic away from √ [−i, −i∞) and φ (ρ) = 0 for all ρ = 0, thus the Riemann surface Σ and the map v → ρ can be analytically extended as long as φ−1 (v) ∩ [−i, −i∞) = ∅. The function φ maps the segments [−i + 0, −i∞ + 0) and √ [−i − 0, −i∞ − 0) onto the lines Re v = −π and Re v = π respectively, hence v → ρ is welldeﬁned and analytic for all v < π. Thus, the map eσ ln(1−iρ) (1 − iρ)σ =λ f : λ → λ ρ ρ √ is holomorphic from the open disk {0 < λ < π} ⊂ Σ to C. Let
f (k) (0) , k = 0, . . . , N − 1, k! and deﬁne rN (λ) so that (3.30) holds. Then rN (λ) satisﬁes ck (σ) =
rN (λ) =
∞ f (k) (0) k λ k!
k=N
28
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
√ for 0 ≤ λ < π. In order to prove (3.31), it is suﬃcient to show that there exists a constant C independent of N such that √ N rN ( v) ≤ Cv 2 , (3.32) v ≥ 0, N ≥ 2, 0 ≤ σ ≤ 1. (k)
Cauchy’s estimates imply that the numbers ck (σ) = f k!(0) are uniformly bounded for all k ≥ 0 and √ σ ∈ [0, 1]. Thus, an application of Cauchy’s estimates in a disk of radius λ = r < π yields N k ∞ λ λ 1 rN (λ) ≤ Mr ≤ Mr , λ < r, r r 1 − λ k=N r √ where Mr = maxλ=r f (λ). Choosing any r ∈ (1, π) we see that such a C exists for 0 ≤ v ≤ 1. On the other hand, equations (3.28) and (3.30) imply that such a C exists also for v ≥ 1. This proves (3.32) and hence also (3.30) and (3.31). Equation (3.30) together with the identity ∞ Γ(α + 1) e−tv v α dv = , α > −1, tα+1 0 (1)
imply the following asymptotic expansion of GU : s N +1 N −1 Γ( 2 ) ck Γ( k+1 t (1) 2 ) e−it + O GU (t, σ; t) = (3.33) . N +1 k+1 2π 2 t 2 −σ t k=0 Proof of (3.20) Equations (3.13), (3.26), and (3.33) give the asymptotics of GL and GU . Substitution into (3.23) yields t [ 2π ]
ζ(1 − s) =
n
s−1
n=1
1 − s
t 2π
s
iπs ∞ N −1 e 2 −nz+it ln z + e (2π)s n=1 j=0 iπs ∞ N −1 e− 2 −nz+it ln z + e (2π)s n=2 j=0
+
t 2π
s
−it
e
1 n− 1 n−
2N ck (σ)Γ( k+1 ) 2
k=0
t
k+1 2
it z
d dz
it z
d dz
j j
z σ−1 n − it z z=−it z σ−1 n − it z z=it
2N σ−N −1 + O (2N + 1)!!N 2 t ,
where we have replaced N by 2N + 1 in (3.33) and used that Γ(N + 1) = N ! ≤ (2N + 1)!! to eliminate one of the error terms. Replacing σ by 1 − σ and taking the complex conjugate of the resulting equation, we ﬁnd (3.20). Proof of (3.22) Letting N = 3 in (3.20) and using the expressions in (3.21) for {ck }60 , we ﬁnd that the term on the rhs of (3.20) involving the ck ’s yields the term involving the second curly bracket on the rhs of (3.22). On the other hand, using that 1 n+
it z
d z −σ z 1−σ (nσz + i(σ − 1)t) =− it dz n + z (nz + it)3
3. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR t ≤ η < ∞
and
1 n+
it z
d dz
2
29
z 1−σ n2 σ(σ + 1)z 2 + in 2σ 2 − σ − 2 tz − (σ − 1)2 t2 z −σ = , (nz + it)5 n + it z
long but straightforward computations show that the two terms involving the double sums on the rhs of (3.20) yield the term involving the ﬁrst curly bracket on the rhs of (3.22). This proves (3.22). Remark 3.3. Using the basic identity (1.2), it is possible to analyze the case of 0 < η < t in a way that is similar to the case of η > t. Actually, this approach is implemented in [9] where the asymptotics of the Hurwitz zeta function to all orders is computed. However, in the next section we ﬁnd it convenient to analyze the case of 0 < η < t by implementing the approach introduced by Siegel. We expect that Siegel’s approach can be used also to obtain asymptotic formulas for the Hurwitz and Lerch zeta functions. In this regard, we note that an analog of the approximate functional equation (1.5) for the Lerch zeta function is derived in [10] and that a RiemannSiegel type integral representation for the Lerch zeta function is presented in [3]. The large t asymptotics of the second moment of the Lerch function is derived in [11]. Another analog of (1.5) for the Lerch function as well as an alternative proof of the large t asymptotics of its second moment can be found in [18].
CHAPTER 4
The Asymptotics of the Riemann Zeta Function for 0 < η < t In this chapter, we consider the asymptotics of ζ(s) as √ t → ∞ with √ 0 < η < t. Theorem 4.1 and its corollary treat the cases < η < t and 2π t < η < 2π t under the assumption that dist(η, 2πZ) >
for some
> 0. The case when η is of √ the same order as t and the case when dist(η, 2πZ) → 0 are covered by theorem 4.4 and its corollary. Throughout this chapter, we assume that the branch cut for the logarithm runs along the positive real axis. Recall the standing assumption (1.24) that η, t ∈ / 2πZ. Theorem 4.1 (The asymptotics to all orders for the case < η < For every > 0, there exists a constant A > 0 such that
√ t).
(4.1) [t]
η
[ 2π ] η 1 1 ζ(s) = + χ(s) ns n1−s n=1 n=1
k+ 12 [ N 2−1 ] ϕ(2k) (0) k 2η 2 1 e−iπs Γ(1 − s) −([ ηt ]+1)iη iπ (s−1) s−1 iπ e i e2 η e4 Γ k+ − 2πi (2k)! t 2 k=0
−iπs
− πt 2
Γ(1 − s)e η ⎧ N ⎪ η 6 ⎪ √ , ⎨O 2N t t × 2 N2+1 ⎪ − At 2 ⎪ η ⎩O N e + Ntη ,
+e
σ−1
1
< η < t 3 < ∞, 1
t3 < η
,
1≤N
0 be given and suppose that 4 < η < t, 0 ≤ σ ≤ 1, m = [t/η], dist(η, 2πZ) > 4 , and N ≥ 1. All error terms of the form O(·) will be uniform with respect to η, σ, N (but not with respect to ). In order to analyze the integral I2 , we write it (w−iη)2 e 2η2 ϕ(w − iη)dw, I2 = e−(m+1)iη (iη)s−1 C2
where 2 z it t t (s−1) ln(1+ iη )− ηt z− 2η 2 z +( η −[ η ])z
ϕ(z) =
e
Deﬁning φ(z) by (4.4)
.
ez − e−iη
2 z it (s−1) ln 1+ iη − ηt z− 2η 2z
φ(z) = e
,
we have φ(z)e( η −[ η ])z . ez − e−iη t
ϕ(z) =
t
We split the contour C2 as follows: C2 = C2 ∪ C2r , where C2 denotes the segment of C2 of length 2 which consists of the points within a distance from iη. We write I2 = I2 + I2r , where I2 = e−(m+1)iη (iη)s−1
it
e 2η2 C2
(w−iη)2
ϕ(w − iη)dw
34
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
and
I2r = e−(m+1)iη (iη)s−1
it
e 2η2
(w−iη)2
C2r
ϕ(w − iη)dw.
We claim that I2r can be estimated as follows: σ 2 πt η − t I2r = O e− 2 √ e 12η2 . (4.5) t iπ
Indeed, the change of variables w = iη + λe 4 gives − η 2 πt iπ − t λ2 I2r = O e− 2 η σ−1 + e 2η2 ϕ(λe 4 )dλ . − η2
We have (4.6)
( t −[ t ])z e η η ez − e−iη = O(1),
iπ
η . 2
< λ
, 0 ≤ σ ≤ 1, t → ∞, η where the error terms are uniform for all η, σ, N within the above ranges, and the function ψ(z) is deﬁned by dist
iηz
ψ(z) =
iη 2
e−s ln(1+ 2πt )+ 8π2 t z ez − e
2πit η
2
η −[ 2π ]z
.
Proof. We replace σ by 1 − σ in (4.1) and take the complex conjugate of both sides. We then multiply the resulting equation by χ(s) and use the identities ζ(¯ s) = ζ(s),
Γ(¯ s) = Γ(s),
χ(s)χ(1 − s) = 1,
(4.25)
χ(¯ s) = χ(s),
χ(s)ζ(1 − s) = ζ(s).
This yields the following equation [t]
ζ(s) = χ(s)
η
η [ 2π ]
1 n1−s
+
n=1 iπ(1−s)
1 ns n=1
Γ(s) ([ ηt ]+1)iη iπs −s − iπ e e 2 η e 4 2πi 2 z it t 2 k+ 12 [ N 2−1 ] d2k e−s ln(1− iη )+ 2η2 z −[ η ]z 1 1 k 2η × Γ k+ (−i) dz 2k z=0 ez − eiη (2k)! t 2
+ χ(s)
e
k=0
+ χ(s)eiπ(1−s) Γ(s)e− 2 η −σ ⎧ 2N N6 η 1 ⎪ ⎪ √ ,
< η < t 3 < ∞, 1 ≤ N < At ⎨O t η3 , t × N +1 √ 2 2 1 ⎪ − At ⎪ ⎩O N e η2 + Ntη , t 3 < η < t < ∞, 1 ≤ N < At η2 , πt
dist(η, 2πZ) > , Replacing η by
2πt η ,
we ﬁnd (4.24).
0 ≤ σ ≤ 1,
t → ∞,
40
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
√ Theorem 4.4 (The asymptotics to all orders for the case t < η < t). For every > 0, there exists a constant A > 0 such that [t]
(4.26)
η
[ 2π ] η 1 1 ζ(s) = + χ(s) ns n1−s n=1 n=1
η 2 2t η it iπ(s−1) [ ]πi−it− 2η 2 (2[ 2π ]π−η) + e−iπs Γ(1 − s) e 2 η s−1 e η 2π SN (s, η)
√
t < η < t,
N πt 3N 6 η σ √ + O e− 2 , t t 0 ≤ σ ≤ 1,
1 ≤ N < At,
t → ∞,
where the error term is uniform for all η, σ, N in the above ranges and the function SN (s, η) is deﬁned by k n n η (4.27) SN (s, η) = an 2 πi − iη k 2π n=0 k=0 2πt 2t 2πt η t 1 n−k − − × ∂2 Φ − 2 , − 2 η η η 2π η 2 N −1
with ∂2 denoting diﬀerentiation with respect to the second argument. The coeﬃcients an are determined by the recurrence formula (4.28)
iη(n + 1)an+1 = (σ − n − 1)an −
it an−2 , η2
n = 0, 1, 2, . . . ,
together with the initial conditions a−2 = a−1 = 0 and a0 = 1, and the function Φ(τ, u) is deﬁned by (4.29)
2
Φ(τ, u) = 01
eπiτ x +2πiux dx, eπix − e−πix
τ < 0,
u ∈ C,
with the contour 0 1 denoting a straight line parallel to e3πi/4 which crosses the real axis between 0 and 1.2 In the particular case when p, q > 0 are integers, we have
(4.30)
2 In
q−1 2p p 1 Φ − ,u = (−1)n e−πin q −2πinu q 1 − (−1)q e−πiqp−2πiqu n=0 p−1 1 2 (−1)q e−πiqp−2πiqu 3πi πiq (u+n+ ) 2 + e 4 e p . p/q n=0
the particular case of η = paper [18, Eq. (1.4)].
√
2πt, an analog of (4.26) is stated without proof in the recent
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
41
For N = 3 equation ( 4.26) simpliﬁes to (4.31) [t]
η
[ 2π ] η 1 1 ζ(s) = + χ(s) s n n1−s n=1 n=1
η 2 2t η it iπ(s−1) [ ]πi−it− 2η 2 (2[ 2π ]π−η) + e−iπs Γ(1 − s) e 2 η s−1 e η 2π σ−1 η πi − iη Φ × Φ+ ∂2 Φ + 2 iη 2π 2 (σ − 2)(σ − 1) 2 η η − πi − iη ∂ πi − iη Φ + 2 2 Φ + 2 Φ ∂ 2 2 2η 2 2π 2π σ √ πt η + O e− 2 ,
t < η < t, 0 ≤ σ ≤ 1, t → ∞, t
where the error term is uniform for all η, σ in the above ranges and Φ and its partial derivatives are evaluated at the point ( 1.8). 2πt Remark 4.5. 1. If η = b where b = p/q > 0 is a rational number, then equations (4.26)(4.30) provide an explicit asymptotic expansion of ζ(s) to all orders. The particular case b = 1 is the famous case analyzed by Siegel in [24] using ideas from Riemann’s unpublished notes. We note that, building on the work of [16], an elementary (but formal) derivation of the RiemannSiegel formula is presented in [5]. 2. The results of theorem 4.4 are convenient for √ studying the higherorder t. However, if the order of η asymptotics of ζ(s) in the case when η = constant × √ is strictly smaller or larger than the order of t, the results of theorem 4.4 are less convenient, because in this case the asymptotics of ζ(s) involves the asymptotics of the sums in (4.30) as p and/or q tend to inﬁnity. In this case, the alternative representation of the asymptotics of theorem 4.1, which avoids the appearance of the above sums, is usually more convenient. 3. The error term in (4.26) is uniform with respect to η, σ, N . More precisely, this means that equation (4.26) is equivalent to the following statement: For every
> 0, there exist constants A > 0, C > 0, K > 0 such that the inequality η [ ηt ] [ 2π ] 1 1 − χ(s) ζ(s) − ns n1−s n=1 n=1 η 2 2t η it iπ(s−1) [ ]πi−it− (2[ ]π−η) 2π 2η 2 − e−iπs Γ(1 − s)e 2 η s−1 e η 2π SN (s, η) N πt 3N 6 η σ √ , ≤ Ce−iπs Γ(1 − s)e− 2 t t holds for all t > K and all σ, N, η such that 0 ≤ σ ≤ 1, 1 ≤ N < At, and √
t < η < t. 4. The main diﬀerence between the proofs of theorems 4.1 and 4.4 can be explained as follows. Consider the representation (4.3) of ζ(s). If η is of strictly smaller order than t1/2 , the main contribution to the asymptotics of ζ(s) comes
42
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
from the part of the contour C2 that lies within a distance of the critical point iη; the remaining part of C2 gives a contribution which is suppressed by a factor −const× ηt2 of the form e cf. Eq. (4.5). Thus, the proof of theorem 4.1 relies on a direct study of the integral near the critical point √ iη using the method of steepest descent. On the other hand, in the case when η ∼ t, the asymptotic expansion of ζ(s) depends on the integral along all of the contour C2 . The contour C2 is a line segment of total length η centered on the critical point iη. In the proof of theorem 4.4, we handle this integral by relating it to the function Φ(τ, u) deﬁned in (4.29). 5. The coeﬃcients an satisfy the estimate [n] t3 an = O , (not uniformly in n); ηn indeed, assuming this estimate up to n, we ﬁnd an+1 = O
n
t[ 3 ] η n+1
+O
n−2
t t[ 3 ] η 3 η n−2
=O
n+1 t[ 3 ] , η n+1
(not uniformly in n).
Proof of Theorem 4.4. Let t > 0, 0 ≤ σ ≤ 1, and m = [t/η]. As in the proof of theorem 4.1, ζ(s) is given by (4.3) where the contributions from I1 , I3 , and I4 are exponentially small, and the error terms of the form O(·) are uniform with respect to the variables η, σ, N as t → ∞. It remains to analyze the integral I2 . In the neighborhood of w = iη we have 2 w − iη 1 w − iη w = (s − 1) − (s − 1) ln + ··· iη iη 2 iη =
t it (w − iη) + 2 (w − iη)2 + · · · . η 2η
Hence we write t
w
e(s−1) ln iη = e η
2 it (w−iη)+ 2η 2 (w−iη)
φ(w − iη),
where φ(z) is deﬁned by (4.4). Deﬁne {an }∞ 0 by φ(z) =
∞
an z n ,
z < η,
n=0
The identity dφ = dz
s−1 t it − − 2 z φ(z) iη + z η η
implies (iη + z)
∞ n=1
nan z
n−1
= s − 1 − (iη + z)
t itz + 2 η η
∞
an z n .
n=0
Hence the coeﬃcients an are determined in succession by the recurrence formula (4.28) supplemented with the conditions a−2 = a−1 = 0 and a0 = 1. The ﬁrst few
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
43
coeﬃcients are given by σ−1 (σ − 2)(σ − 1) , a2 = − , iη 2η 2 −2t + i(σ − 3)(σ − 2)(σ − 1) a3 = , 6η 3 (σ − 4)(σ − 3)(σ − 2)(σ − 1) + 2i(4σ − 7)t a4 = . 24η 4
a0 = 1,
a1 =
Representing φ(z) in the form φ(z) =
N −1
an z n + rN (z),
n=0
we ﬁnd
φ(w) zN dw, z < η, N 2πi Γ w (w − z) where Γ is a contour contained in the disk of radius η centered at the origin which encircles the points 0 and z once. 3 Let z < 47 η (so that 21 20 z < 5 η) and let Γ be a circle with center w = 0 and radius ρN , where 21 3 z ≤ ρN ≤ η. 20 5 Equation (4.7) implies the following analog of (4.8): rN (z) =
Re ln φ(z) ≤ σ − 1 ln Hence rN (z) = O
3
5tρN 2πρN 6η 3 zN ρ−N N e ρN − z
8 5tz3 + , 5 6η 3
z ≤
3 η. 5
5tρ3 N N −N 6η3 = O z ρN e ,
z
0 such that (4.39) I2S =
t
C2
(iη)s−1
eη
(w−iη)+ 2i
t η2
−1 (w−iη)2 −mw N
ew − 1
an (w − iη)n dw + O(e−
πt 2 −At
),
n=0
where C2 denotes the inﬁnite straight line of which C2 is a part. Indeed, in view of (4.38), if we replace C2 by C2 , the integral multiplying an in the expression for I2S changes by O η σ−1 e−
(4.40)
πt 2
∞
− 2ηt 2 λ2 n
e
λ dλ .
η 2
We write the integrand as − 4ηt 2 λ2 n
− 4ηt 2 λ2
λ ×e
e
.
2n t η,
and so it decreases throughout The ﬁrst factor is steadily decreasing for λ > the interval of integration provided that n < N < At, with A suﬃciently small. The term in (4.40) is then (4.41) O η
− 4ηt 2 σ−1 − πt 2
e
e
η2 4
η n 2
∞
− 4ηt 2 λ2
e
dλ
η 2
Also, by (4.32), choosing z such that z ≤
2 πt − t η η n η √ . = O η σ−1 e− 2 e 4η2 4 2 t
20 2 13 21 ( 5t ) η,
we ﬁnd
−n
(4.42)
an = (rn (z) − rn+1 (z))z n n+1 3 5et 3 5et n n+1 −n =O z − z z 2nη 3 2(n + 1)η 3 n 5et 3 =O , 1 ≤ n ≤ N − 1. 2nη 3
Multiplying (4.41) by an and summing from 0 to N − 1, we ﬁnd that the total error is n N −1 n t 5et 3 η η σ−1 − πt − e 2 16 √ a0 + O η 2 2nη 3 t n=1 n N −1 t 5et 3 η σ−1 − πt − e 2 16 √ 1 + . =O η 16n t n=1
46
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t n
Now the factor (t/n) 3 increases steadily up to n = t/e, and so if n < At, where A < 1/e, it is 1 1 O(e 3 tA ln A ). Hence if N < At, with A suﬃciently small, the total error is O(e−
πt 2 −At
).
This proves (4.39). Finally, we analyze the sum 2 t it N −1 (w−iη)+ 2η 2 (w−iη) −mw eη (iη)s−1 (w − iη)n dw. (4.43) an w −1 e C2 n=0 The integral in (4.43) may be expressed as η η η 2 t it t (w + 2[ 2π ]πi − iη)n (w+2[ 2π ]πi−iη)+ 2η 2 (w+2[ 2π ]πi−iη) −[ η ]w dw, − eη w e −1 L where L is a line in the direction arg w = π/4, passing between 0 and 2πi. This is n! times the coeﬃcient of ξ n in the following expression: (4.44) η η η 2 t it t dw (w+2[ 2π ]πi−iη)+ 2η 2 (w+2[ 2π ]πi−iη) −[ η ]w+ξ(w+2[ 2π ]πi−iη) eη − w −1 e L η η 2 t it it 2πt η t dw 2[ η ]πi−it+ 2η w2 +( 2t 2 (2[ 2π ]πi−iη) +ξ(2[ 2π ]πi−iη) η − η 2 [ 2π ]−[ η ]+ξ)w = −e η 2π e 2η2 w −1 e L η η 2 t it 1 2πt 2t 2πt η t η 2[ 2π ]πi−it+ 2η 2 (2[ 2π ]πi−iη) =e − +ξ− 2πiΦ − 2 , − 2 η η η 2π η 2 η
× eξ(2[ 2π ]πi−iη) , where the function Φ(τ, u) is deﬁned by (4.29), i.e., 2 2 iτ 1 eπiτ x +2πiux e− 4π w +(u+ 2 )w 1 dw, dx = Φ(τ, u) = πix − e−πix 2πi −L ew − 1 01 e
τ < 0,
u ∈ C.
We rewrite the expression on the rhs of (4.44) as ∞ η 2 t it 2πt 2t 2πt η t 1 ξ l 2[ η ]πi−it+ 2η 2 (2[ 2π ]πi−iη) e η 2π 2πi ∂2l Φ − 2 , − 2 − − η η η 2π η 2 l! l=0
×
∞ ξ k (2[
η 2π ]πi
− iη)k
k!
k=0
.
It follows that the expression in (4.43) is given by t
(4.45)
2πi(iη)s−1 e η
η η 2 it 2[ 2π ]πi−it+ 2η 2 (2[ 2π ]πi−iη)
SN (s, η)
where SN (s, η) is deﬁned in (4.27). Equations (4.37), (4.39), and (4.45) show that I2 satisﬁes t
2[
η
]πi−it+
it
(2[
η
]πi−iη)2
2π 2η 2 I2 = 2πi(iη)s−1 e η 2π SN (s, η) N6 σ πt πt 3N η √ + O(e− 2 −At ). + O e− 2 t t Equation (4.26) follows by substituting this expression into (4.3).
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
47
In order to prove (4.30) we suppose that u ∈ C and τ < 0. We claim that Φ satisﬁes the two recursion relations 3πi
πi 1 2 e 4 Φ(τ, u) = Φ(τ, u + 1) − e− τ (u+ 2 ) τ 
(4.46) and
Φ(τ, u) = 1 − eπiτ −2πiu Φ(τ, u − τ ).
(4.47)
Repetitive use of these two equations yields the following identities: 3πi N −1 e 4 − πi (u+n+ 1 )2 2 e τ Φ(τ, u) = Φ(τ, u + N ) − τ  n=0
(4.48) and (4.49)
Φ(τ, u) =
N −1
2
(−1)n eπin
τ −2πinu
+ (−1)N eπiN
2
τ −2πiN u
Φ(τ, u − N τ )
n=0
valid for all N ≥ 0. If τ = −p/q, we apply (4.48) with N = p and (4.49) with N = q to ﬁnd 3πi p−1 e 4 − πi (u+n+ 1 )2 2 Φ(τ, u) = Φ(τ, u + p) − e τ τ  n=0
(4.50) and (4.51)
Φ(τ, u) =
q−1
2
(−1)n eπin
τ −2πinu
+ (−1)q eπiq
2
τ −2πiqu
Φ(τ, u + p).
n=0
Eliminating Φ(τ, u + p) from these two equations, we ﬁnd (4.30). It remains to prove (4.46) and (4.47). In order to prove (4.46), we note that
2
Φ(τ, u + 1) − Φ(τ, u) =
eπiτ x 01
2
eπiτ x
= 01
1 2 − πi τ (u+ 2 )
e2πi(u+1)x − e2πiux dx eπix − e−πix +2πi(u+ 12 )x
dx
eπiτ (x+
=e
u+ 1 2 τ
01
= e− τ
πi
(u+ 12 )2
2
eπiτ x dx. 01
The identities
2
eπiτ x dx = 01
imply equation (4.46).
3πi
2 dy e 4 = e−πiy τ  τ  01
)2
dx
48
4. THE ASYMPTOTICS OF THE RIEMANN ZETA FUNCTION FOR 0 < η < t
The identity (4.47) is established as follows: 2 eπiτ x +2πiux dx Φ(τ, u) = πix − e−πix 01 e 2 eπiτ x +2πiux =1+ dx πix − e−πix −10 e 2 eπiτ (x−1) +2πiu(x−1) =1+ dx πi(x−1) − e−πi(x−1) 01 e 2 eπiτ x +2πi(u−τ )x πiτ −2πiu =1−e dx eπix − e−πix 01 = 1 − eπiτ −2πiu Φ(τ, u − τ ), where the second equality follows from the fact that the residue of the integrand at 1 . x = 0 is 2πi √ Theorem 4.4 is valid for
t < η < t. As a corollary, we can ﬁnd an analogous √ t result valid for 2π < η < . Corollary 4.6 (The asymptotics to all orders for the case 2π < η < For every > 0, there exists a constant A > 0 such that (4.52) [t]
√ t ).
η
[ 2π ] η 1 1 ζ(s) = + χ(s) ns n1−s n=1 n=1
s iη 2 t t t 2 iπs η 2πt −iη[ ]+it+ ([ ]− ) iπ(1−s) η 2t η η + χ(s)e Γ(s) e 2 e SN 1 − s¯, 2πt η N6 2πt 1−σ ( η ) 3N πt √ + O e− 2 , t t √ t , 0 ≤ σ ≤ 1, 1 ≤ N < At, t → ∞, 2π < η <
where the error term is uniform for all η, σ, N in the above ranges and SN is given by ( 4.27).
Proof. We replace σ by 1 − σ in (4.26) and take the complex conjugate of both sides. We then multiply the resulting equation by χ(s) and use the identities (4.25). This yields the following equation [t]
η [ 2π ]
1 ns n=1 n=1
η 2 it iπs − 2t [ η ]πi+it+ 2η 2 (2[ 2π ]π−η) + χ(s)eiπ(1−s) Γ(s) (e 2 η −s e η 2π SN (1 − s¯, η)
ζ(s) = χ(s)
η
1
n1−s
+
√
t < η < t, Replacing η by
2πt η ,
N 3N 6 η 1−σ − πt √ +O e 2 , t t 0 ≤ σ ≤ 1,
we ﬁnd (4.52).
1 ≤ N < At,
t → ∞.
CHAPTER 5
Consequences of the Asymptotic Formulae Using theorems 3.1 and 3.2 we can compute several interesting sums. Theorem 5.1. Deﬁne the polylogarithm Lim (z) by ( 3.3). The following relation holds: (5.1) η2 [ 2π ] 1 η2 1−s η1 1−s n−s = − 1 − s 2π 2π η1 n=[ 2π ]+1
iη1 j iπ(1−s) ∞ N −1 e− 2 d z −σ 1 −nz−it ln z + e (2π)1−s n=1 j=0 dz n + it n + it z z z=iη2 j −iη1 iπ(1−s) ∞ N −1 d z −σ e 2 1 −nz−it ln z + e (2π)1−s n=1 j=0 dz n + it n + it z z z=−iη2 1 + 2(N +1) + O (2N + 1)!!N η1−σ−N ,
(1 + )t < η1 < η2 < ∞, > 0, 0 ≤ σ ≤ 1, N ≥ 2, t → ∞,
where the error term is uniform for all η1 , η2 , , σ, N in the above ranges. For N = 3 equation ( 5.1) simpliﬁes to (5.2) η
2] [ 2π
η
1 ]+1 n=[ 2π
n
−s
1 η2 1−s η1 1−s = − 1 − s 2π 2π
2iη1−s t − iσ + Re Li2 (eiη1 ) −i arg(1 − eiη1 ) + (2π)1−s η1 ' 1& + 2 it2 − 3iσt − (σ − 1)t − iσ(σ + 1) Im Li3 (eiη1 ) η1 −s 2iη2 t − iσ − Re Li2 (eiη2 ) −i arg(1 − eiη2 ) + (2π)1−s η2 ' 1& 2 iη2 + 2 it − 3iσt − (σ − 1)t − iσ(σ + 1) Im Li3 (e ) η2 1 + 8 1 3 + O 3+σ t + ,
η1 (1 + )t < η1 < η2 < ∞, > 0, 0 ≤ σ ≤ 1, t → ∞,
where the error term is uniform for all η1 , η2 , , σ in the above ranges. 49
50
5. CONSEQUENCES OF THE ASYMPTOTIC FORMULAE
Proof. Replacing in equation (3.1) η with η1 and subtracting the resulting equation from the equation obtained from (3.1) by replacing η with η2 , we ﬁnd (5.1). Equation (5.2) follows in a similar way from (3.2). Theorem 5.2. Deﬁne the polylogarithm Lim (z) by ( 3.3). The following relation holds: (5.3) η 1−s [ 2π ] t 1 η 1−s n−s = − 1 − s 2π 2π t n=[ 2π ]+1
it j iπ(1−s) ∞ N −1 d z −σ e− 2 1 −nz−it ln z + e (2π)1−s n=1 j=0 dz n + it n + it z z z=iη j iπ(1−s) ∞ N −1 d z −σ e 2 1 −nz−it ln z − e it (2π)1−s n=1 j=0 n + z dz n + it z z=−iη j iπ(1−s) ∞ N −1 d z −σ e 2 1 −nz−it ln z + e (2π)1−s n=2 j=0 dz n + it n + it z z z=−it 1−s 2N ck (1 − σ)Γ( k+1 t 2 ) + eit k+1 2π t 2 k=0 2(N +1) (2N + 1)!!N ( 1+ (2N + 1)!!N 22N ) +O + , tσ+N η σ+N (1 + )t < η < ∞, > 0, 0 ≤ σ ≤ 1, N ≥ 2, t → ∞,
where the error term is uniform for all η, , σ, N in the above ranges and the coefﬁcients ck (σ) are deﬁned in ( 3.30). For N = 3 equation ( 5.3) simpliﬁes to (5.4) η [ 2π ]
t n=[ 2π ]+1
n
−s
1−s ! t 1 η 1−s = − + ··· 1 − s 2π 2π 1 + 8 1 1 + O 3+σ t3 + + 3+σ , η
t (1 + )t < η < ∞, > 0,
0 ≤ σ ≤ 1,
t → ∞,
where the error term is uniform for all η, , σ in the above ranges and {· · · } denotes an explicit expression involving σ, t, and the polylogarithms Lim (eit ), m = 1, . . . , 5. Proof. Equation (5.3) follows by subtracting equation (3.20) from equation (3.1). Similarly, equation (5.4) follows by subtracting equation (3.22) from equation (3.2). The following equation is given on page 78 of Titchmarsh [26]: 1 1 (5.5) ∼ χ(s) . s 1−s n n t t x 0 such that (5.6) [ ηt ]
η
1
n=[ ηt ]+1
1 = χ(s) ns
2] [ 2π
1 n1−s
η
1 ]+1 n=[ 2π
η 2 2t η it iπ(s−1) [ ]πi−it− 2η 2 (2[ 2π ]π−η) + e−iπs Γ(1 − s) e 2 η s−1 e η 2π SN (s, η) η2 N 3N 6 η σ − πt √ +O e 2 , t t η=η1 √
t < η1 < η2 < t, 0 ≤ σ ≤ 1, 1 ≤ N < At, t → ∞,
2
where the error term is uniform for all η1 , η2 , σ, N in the above ranges and the function SN (s, η) is deﬁned by ( 4.27). For N = 3 equation ( 5.6) simpliﬁes to [ ηt ]
(5.7)
1
n=[ ηt ]+1 2
η
1 = χ(s) ns
2] [ 2π
η
1 ]+1 n=[ 2π
1 n1−s
η 2 t it iπ(s−1) 2[ η ]πi−it− 2η 2 (2[ 2π ]π−η) Γ(1 − s) e 2 η s−1 e η 2π +e σ−1 η πi − iη Φ × Φ+ ∂2 Φ + 2 iη 2π (σ − 2)(σ − 1) 2 η − πi − iη ∂2 Φ ∂2 Φ + 2 2 2η 2 2π 2 η2 ησ η − πt πi − iη Φ + O e 2 + 2 , 2π t η=η1 √
t < η1 < η2 < t, 0 ≤ σ ≤ 1, t → ∞, −iπs
where the error term is uniform for all η1 , η2 , σ in the above ranges, Φ is deﬁned by ( 4.29), and Φ and its partial derivatives are evaluated at the point ( 1.8). Proof. Replacing in equation (4.26) η with η1 and subtracting the resulting equation from the equation obtained from (4.26) by replacing η with η2 , we ﬁnd (5.6). Equation (5.7) follows in a similar way from (4.31). Remark 5.4. The relation (5.5) is a particular case of (5.6) (let x = t/η2 and N = t/η1 ).
Part 2
Asymptotics to all Orders of a TwoParameter Generalization of the Riemann Zeta Function
CHAPTER 6
An Exact Representation for Φ(u, v, β) The twoparameter generalization Φ(u, v, β) of ζ(s) was deﬁned in (1.12). In Theorem 2.1, we derived an exact representation for ζ(s). In this chapter, we prove Theorem 6.1 which provides an analogous representation for Φ(u, v, β). Recall that we make the basic assumption (1.24); in particular η ∈ / 2πZ. The branch cut for the logarithm is assumed to run along the negative real axis. Theorem 6.1 (An exact representation for Φ). Let Φ(u, v, β) be deﬁned by ( 1.12). Then −iπu
Φ(u, v, β) = (e
−iπv
− 1)e
−iπv iπu +e (1 − e )
η [ 2π ] iπ 2
(2πe )
u+v−1
mu−1 (m + β)v−1
m=1
dz +(e − 1) z u−1 (z + 2iπβ)v−1 z e −1 −iη iη ' z u−1 & (z − 2iπβ)v−1 + e−iπv e−z (z + 2iπβ)v−1 dz + −z − 1 L3 e ' z u−1 & (z − 2iπβ)v−1 + e−iπv e−z (z + 2iπβ)v−1 dz, + (e−iπu − 1) −z −1 ˆα e C η ∞eiφ1
−iπu
∞eiφ2
(6.1) u, v ∈ C,
β > 0,
0 < η < 2πβ,
−
π π < φj < , j = 1, 2, 2 2
where 0 < α < 2π min(1, β) and the contours L3 and Cˆηα with the orientations shown in ﬁgures 2.1 and 2.3 are deﬁned in ( 2.5) and ( 2.10c), respectively. Proof. We decompose the contour Hα in the deﬁnition (1.12) of Φ into the union of the three contours {Lj }31 deﬁned in (2.5) with the orientation shown in ﬁgure 2.1: Hα = L 1 ∪ L 2 ∪ L 3 . Proceeding in analogy with the proof of Theorem 2.1, we write the integral along L1 as follows: ∞eiπ dz dz u−1 v−1 = z (z − 2iπβ) z u−1 (z − 2iπβ)v−1 −z −z − 1 e e −1 L1 iα ∞ dζ ζ u−1 (−ζ − 2iπβ)v−1 ζ = eiπu e −1 −iα ∞ dζ = eiπu e−iπ(v−1) (6.2) , ζ u−1 (ζ + 2iπβ)v−1 ζ e −1 −iα 55
56
6. AN EXACT REPRESENTATION FOR Φ(u, v, β)
iη Cˆηα
Cαη iα Re z
−iα −α C−η
−iη
−α Figure 6.1. The contours Cαη , C−η and Cˆηα .
where the ﬁrst equality is a consequence of Cauchy’s theorem, the second equality is a consequence of the substitution z = eiπ ζ, and the third equality uses the fact that −ζ − 2iπβ is in the third quadrant. Similarly, the integral along L2 can be written in the form (6.3) z u−1 (z − 2iπβ)v−1 L2
dz = e−iπu e−iπ(v−1) e−z − 1
iα
ζ u−1 (ζ + 2iπβ)v−1 ∞
dζ . eζ − 1
The starting point for deriving this identity is the application of Cauchy’s theorem in the domain enclosed by L2 and by the ray from ∞e−iπ to −iα, i.e., in the shaded domain 2 of ﬁgure 2.2. The integral along L3 can be written as follows: dz dζ = −e−iπ(v−1) z u−1 (z − 2iπβ)v−1 −z ζ u−1 (ζ + 2iπβ)v−1 ζ e −1 e −1 L3 L3 v−1 (ζ − 2iπβ)v−1 −iπ(v−1) (ζ + 2iπβ) + e ζ u−1 + dζ e−ζ − 1 eζ − 1 L3 ∞ iα dζ −iπ(v−1) + ζ u−1 (ζ + 2iπβ)v−1 ζ =−e e −1 −iα ∞ ' ζ u−1 & (ζ − 2iπβ)v−1 + e−iπv e−ζ (ζ + 2iπβ)v−1 dζ, + (6.4) −ζ −1 L3 e where the ﬁrst equality is an identity and the second equality follows from an application of Cauchy’s theorem in the domain enclosed by L3 and the two rays (−iα, ∞) and (∞, iα), i.e., in the shaded domain 3 of ﬁgure 2.2. Adding equations (6.2)(6.4) we obtain ∞ πu iπu ∞ dz −iπv − iπu 2 2 sin +e Φ(u, v, β) = − 2ie z u−1 (z + 2iπβ)v−1 z e 2 e −1 −iα iα u−1 & ' z (6.5) (z − 2iπβ)v−1 + e−iπv e−z (z + 2iπβ)v−1 dz. + −z −1 L3 e −α ˆ α Deﬁning the contours Cαη , C−η , Cη as in (2.10) with the orientations shown in
6. AN EXACT REPRESENTATION FOR Φ(u, v, β)
57
ﬁgure 6.1, the second integral in the rhs of (6.5) can be rewritten as the sum of an integral along the contour Cαη plus an integral along the ray (iη, ∞eiφ1 ). Similarly, the ﬁrst integral in the rhs of (6.5) can be rewritten as the sum of an integral along −α the curve −C−η plus an integral along the ray (−iη, ∞eiφ2 ). Hence the ﬁrst two terms in the rhs of (6.5) yield the second line in the rhs of (6.1), as well as the additional term I deﬁned by (6.6) πu iπu dz −iπv − iπu 2 2 . I = −2ie sin +e z u−1 (z + 2iπβ)v−1 z −e η −α 2 e −1 C−η Cα −α Letting z = ζe−iπ in the integral involving C−η and using the fact that −ζ + 2iπβ is in the ﬁrst quadrant we ﬁnd (6.7) iπu dz u−1 v−1 dz − iπu iπv 2 2 =e . z (z + 2iπβ) e z u−1 (z − 2iπβ)v−1 −z −e z −1 −α e e −1 ˆα C C−η η
Using the above equation in the rhs of (6.6) and then adding and subtracting in the resulting equation the term πu iπu dz e− 2 e−iπv , z u−1 (z + 2iπβ)v−1 z −2i sin 2 e −1 ˆα C η we ﬁnd that I is given by πu iπu dz − 2 −iπv I = − 2i sin z u−1 (z + 2iπβ)v−1 z e e η 2 e −1 α ˆ Cα ∪Cη (z − 2iπβ)v−1 e−iπv (z + 2iπβ)v−1 − z u−1 + . −z e −1 ez − 1 ˆα C η That is, −iπu
I = (e
(6.8)
+ ˆα C η
−iπv − 1) e
z u−1 (z + 2iπβ)v−1
η ˆα Cα ∪C η
dz ez − 1
' z u−1 & v−1 −iπv −z v−1 (z − 2iπβ) + e e (z + 2iπβ) . e−z − 1
Cauchy’s theorem implies that the ﬁrst integral in the curly bracket in (6.8) equals η [ 2π ] iπ 2
(2πe )u+v−1
mu−1 (m + β)v−1 .
m=1
Thus (6.5) with the aid of equation (6.8) becomes equation (6.1).
CHAPTER 7
The Asymptotics of Φ(u, v, β) √ In Theorem 4.1, we established an asymptotic formula for ζ(s) for < η < t. In this chapter, we establish an analogous asymptotic formula for Φ. Let u = σ1 + it and v = σ2 − it. The function Φ(u, v, β) was deﬁned in (1.12) by dz (7.1) Φ(u, v, β) = , u, v ∈ C, β ∈ R \ {0}, z u−1 (z − 2iπβ)v−1 −z e −1 Hα where Hα , 0 < α < 2π min(1, β), denotes the Hankel contour (1.13) surrounding the negative real axis in the counterclockwise direction, and the complex powers are deﬁned by wa = ea(ln w+i arg w) with arg w ∈ (−π, π], i.e., the branch cut runs along the negative real axis. In this and the following chapter, we prefer to have the branch cut along the positive real axis. Therefore we change variables z = −w in (7.1) to get (7.2) Φ(u, v, β) = −e−iπ(u+v)
wu−1 (w + 2iπβ)v−1 ˆα H
dw , −1
ew
u, v ∈ C, β ∈ R \ {0},
ˆ α denotes the Hankel contour surrounding the positive real axis in the where H counterclockwise direction, i.e., ˆ α = {r + i0  α < r < ∞} ∪ αeiθ  0 < θ < 2π ∪ {r − i0  α < r < ∞} , (7.3) H and the complex powers are deﬁned by wa = ea(ln w+i arg w) with arg w ∈ [0, 2π), i.e., the branch cut now runs along the positive real axis. Using the identity m 1 e−mw = , e−nw + w w e − 1 n=1 e −1
m = 1, 2, . . . ,
we may write Φ(u, v, β) = −e−iπ(u+v)
m n=1
(7.4)
wu−1 (w + 2iπβ)v−1 e−nw dw
ˆα H
+ ˆα H
wu−1 (w + 2iπβ)v−1 e−mw dw . ew − 1
We are interested in the asymptotic behavior of Φ(u, v, β) as t → ∞. Thus we note that the second integral on the rhs of (7.4) has critical points at the solutions of d ln wu−1 (w + 2iπβ)v−1 e−mw = 0, dw 59
60
7. THE ASYMPTOTICS OF Φ(u, v, β)
that is, at
(−2iπβm + u + v − 2)2 + 8iπβm(u − 1) . w= 2m Anticipating that m will grow like tα , α > 0, we ﬁnd that for large t the critical points are approximately given by ( 2t iβπ − 1 ± 1 + . βπm By performing a steepest descent analysis we can ﬁnd the asymptotics of Φ to all orders. The idea is that given some η > 0, we choose m so that the critical point for large t lies at iη. Solving the equation ( 2t η = βπ − 1 + 1 + βπm for m we ﬁnd that m should be given by 1 1 − . t η η + 2πβ But since m has to be an integer, we instead use the approximate deﬁnition 1 1 − m = [x] where x = t . η η + 2πβ √ Theorem 7.1 (The asymptotics of Φ(u, v, β) to all orders for < η < t). For every > 0, there exists a constant A > 0 such that m − eiπ(σ1 +σ2 ) Φ(u, v, β) = wu−1 (w + 2iπβ)v−1 e−nw dw −2iπβm + u + v − 2 ±
n=1
+ i(2π)σ1 +σ2 −1 e 2
πi
ˆα H
η [ 2π ]
(σ1 +σ2 )
nu−1 (n + β)v−1
n=1
− e−(m+1)iη (iη)u−1 (iη + 2πiβ)v−1 e 4
iπ
−k− 12 ϕ(2k) (0) t 1 1 1 k i × − Γ k+ (2k)! 2 η2 (η + 2πβ)2 2 [ N 2−1 ]
k=0
+ η σ1 −1 (η + 2πβ)σ2 −1 ⎧ 22N N6 η ⎪ ⎪ √ , ⎨O t t × 3t − N2+1 ⎪ − At 2 ⎪ η + 4N η2 , ⎩O N e (7.5) β > 0,
< η < 2πβ − < ∞,
1
< η < t 3 < ∞, 1
t3 < η
,
1≤N
8 , and N ≥ 1 (since > 0 is arbitrary, we can replace 8 with at the end). All error terms of the form O(·) will be uniform with respect to η, σ1 , σ2 , β, N in the given ranges, but not with respect to . We let A > 0 denote a generic constant which can change within a computation. Let 1 1 − x=t , m = [x]. η η + 2πβ √ Since η < t and η < 2πβ, we have √ t 1 t x= ≥ . η η 2πβ +1 2 ˆ α into the In particular, m → ∞ as t → ∞. We deform the Hankel contour H straight lines Cj , j = 1, . . . , 4 joining ∞, cη + iη(1 + c), −cη + iη(1 − c), −cη − iη, ∞, where 0 < c ≤ 1/2 is an absolute constant, see ﬁgure 4.1. Then (7.6) η [ 2π 4 ] wu−1 (w + 2iπβ)v−1 e−mw u−1 v−1 dw = − 2πi (2πin) (2πi(n + β)) + Ij , ew − 1 ˆα H η j=1 n=−[ 2π ] n =0
where
Ij = Cj
wu−1 (w + 2iπβ)v−1 e−mw dw, ew − 1
j = 1, . . . , 4.
Note that η [ 2π ]
−2πi
(2πin)u−1 (2πi(n + β))v−1 = −(2π)σ1 +σ2 −1 e
η [ 2π ] iπ 2 (u+v−1)
η n=−[ 2π ]
nu−1 (n + β)v−1
n=1
n =0
− (2π)
σ1 +σ2 −1
η [ 2π ]
e
πi 2 v
e
3πi 2 (u−1)
nu−1 (−n + β)v−1
n=1
= i(2π)σ1 +σ2 −1 e 2
πi
η [ 2π ]
(σ1 +σ2 )
& ' nu−1 (n + β)v−1 − eπiu (−n + β)v−1
n=1
= i(2π)
σ1 +σ2 −1
η [ 2π ]
e
πi 2 (σ1 +σ2 )
nu−1 (n + β)v−1 + O(e−At ).
n=1
In view of (7.4) and (7.6) this gives the ﬁrst two terms on the rhs of (7.5). It remains to analyze the contributions from the Ij ’s. We ﬁrst prove that I1 , I3 , I4 are exponentially small as t → ∞. Let w = ρeiφ , ˜ 0 < φ < 2π, and w + 2iπβ = ρ˜eiφ , 0 < φ˜ < 2π. Then wu−1  = ρσ1 −1 e−tφ ,
˜
(w + 2iπβ)v−1  = ρ˜σ2 −1 etφ .
Also, xη =
tQ , 1+Q
where
Q :=
2πβ > 1. η
62
7. THE ASYMPTOTICS OF Φ(u, v, β)
Analysis of the integral I4 For w ∈ C4 we have ew − 1 > A, and
ρ ≥ η,
ρ˜ ≥ 2πβ − η,
φ ≥ π + arctan
η , cη
2πβ − η Q−1 φ˜ ≤ π − arctan = π − arctan . cη c
Hence ˜
wu−1 (w + 2iπβ)v−1  = ρσ1 −1 ρ˜σ2 −1 e−t(φ−φ) ≤ η σ1 −1 (2πβ − η)σ2 −1 e−t(arctan c +arctan 1
= O(e−t(arctan c +arctan 1
(7.7)
Q−1 c )
Q−1 c )
).
Thus,
wu−1 (w + 2iπβ)v−1 e−mw dw ew − 1 C4 ∞ −t(arctan 1c +arctan Q−1 ) −mw 1 c =O e e dw1
I4 =
−cη
−t(arctan
= O(e
= O(e−t(arctan
Q−1 1 c +arctan c )+mcη Q−1 1 c +arctan c )+xcη
where F (Q) = arctan Since
c2 4
) = O(e−tF (Q) ),
Q−1 cQ 1 + arctan − . c c 1+Q
< Q, we have F (Q) =
Hence
)
(Q +
4cQ − c3 > 0, (c2 + (Q − 1)2 )
1)2
Q > 1.
1 c inf F (Q) ≥ F (1) = arctan − > A > 0. Q>1 c 2
It follows that I4 = O(e−At ). Analysis of the integral I3 For w = w1 + iw2 ∈ C3 we have φ = π − arctan
w2 , cη
w2 + 2πβ φ˜ = π − arctan . cη
The function φ− φ˜ assumes its maximum at w2 = −πβ and decreases symmetrically as w2 moves away from this point. Its minimum on C3 is therefore assumed at the upper endpoint where w2 = (1 − c)η. For w2 = (1 − c)η we have (7.8) w2 w2 + 2πβ − arctan = arctan φ − φ˜ = arctan cη cη
1−c+Q c
− arctan
1−c . c
7. THE ASYMPTOTICS OF Φ(u, v, β)
63
Hence
1−c+Q 1−c wu−1 (w + 2iπβ)v−1 e−mw = O (cη)σ1 −1 (cη)σ2 −1 e−t(arctan( c )−arctan c ) emcη 1−c+Q 1−c = O e−t(arctan( c )−arctan c ) excη = O e−tF (Q) ,
where F (Q) = arctan
1−c+Q c
− arctan
1−c c
−
cQ . 1+Q
Since c < 1 + Q, we have F (Q) = −
2c2 (c − Q − 1) > 0, (Q + 1)2 (c2 + (1 + Q − c)2 )
Hence
inf F (Q) ≥ F (1) = arctan
Q>1
2−c c
− arctan
1−c c
Q > 1. −
c > A > 0. 2
It follows that I3 = O(e−At ). Analysis of the integral I1 For w ∈ C1 we have w = w1 + i(1 + c)η and w1 ≥ cη. Also, ρ ≥ η,
ρ˜ ≥ η + 2πβ,
1+c (1 + c)η = arctan , w1 P 1+c+Q (1 + c)η + 2πβ (7.9) , = arctan φ˜ = arctan w1 P where P := w1 /η ≥ c. There exists A > 0 such that φ = arctan
1 ≤ (1 − A)ec ≤ (1 − A)ew1 , and so ew − 1 ≥ ew1 − 1 ≥ Aew1 . Thus wu−1 (w + 2iπβ)v−1 e−mw ˜ = O η σ1 −1 (η + 2πβ)σ2 −1 e−t(φ−φ)−(m+1)w1 w e −1 ˜ = O η σ1 −1 (η + 2πβ)σ2 −1 e−t(φ−φ)−xw1 = O η σ1 −1 (η + 2πβ)σ2 −1 e−tF (P,Q) , where F (P, Q) = arctan Now
1+c 1+c+Q P − arctan +P − . P P 1+Q
∂F 1 P > 0, − = (c+Q+1)2 ∂Q (Q + 1)2 P + 1 2 P
P ≥ c,
Q ≥ 1,
so F assumes its minimum over the domain {P ≥ c, Q ≥ 1} on the boundary where Q = 1. For deﬁniteness, let us henceforth set 1 c= . 8 Then P ] ∂[F (P, 1) − 16 c > 0 for P ≥ c, and F (c, 1) − > 0, ∂P 16
64
7. THE ASYMPTOTICS OF Φ(u, v, β)
so that F (P, 1) > P/16 for P ≥ c. Hence wu−1 (w + 2iπβ)v−1 e−mw dw I1 = ew − 1 C1 ∞ = O η σ1 −1 (η + 2πβ)σ2 −1 e−tP/16 dw1 cη ∞ σ1 σ2 −1 −tP/16 = O η (η + 2πβ) e dP c = O η σ1 (η + 2πβ)σ2 −1 t−1 e−ct/16 . It follows that I1 = O(e−At ). Analysis of the integral I2 It remains to analyze the integral I2 . We write it 1 ( 1 − )(w−iη)2 −(m+1)iη u−1 v−1 I2 = e (iη) (iη + 2πiβ) e 2 η2 (η+2πβ)2 ϕ(w − iη)dw, C2
where 2 z z 1 1 (u−1) ln(1+ iη )+(v−1) ln(1+ iη+2πiβ )−xz− it 2 ( η 2 − (η+2πβ)2 )z +(x−m)z
ϕ(z) =
e
Deﬁning φ(z) by (7.10)
.
ez − e−iη
2 z z 1 1 (u−1) ln 1+ iη +(v−1) ln(1+ iη+2πiβ )−xz− it 2 ( η 2 − (η+2πβ)2 )z
φ(z) = e
,
we have ϕ(z) =
φ(z)e(x−m)z . ez − e−iη
We split the contour C2 as follows: C2 = C2 ∪ C2r , where C2 denotes the segment of C2 of length 2 which consists of the points within a distance from iη. We write I2 = I2 + I2r , where I2 = e−(m+1)iη (iη)u−1 (iη + 2πiβ)v−1 and I2r = e−(m+1)iη (iη)u−1 (iη + 2πiβ)v−1
it
e2
2 1 ( η12 − (η+2πβ) 2 )(w−iη)
C2
it
e2
2 1 ( η12 − (η+2πβ) 2 )(w−iη)
C2r
ϕ(w − iη)dw
ϕ(w − iη)dw.
We claim that I2r can be estimated as follows: σ1 t2 η (η + 2πβ)σ2 −1 − 32η r 2 √ I2 = O e (7.11) . t iπ
Indeed, the change of variables w = iη + λe 4 gives − √2cη 2 1 iπ − 2t ( η12 − (η+2πβ) r σ1 −1 σ2 −1 2 )λ 4 I2 = O η (η + 2πβ) + ϕ(λe )dλ . e √ − 2cη
7. THE ASYMPTOTICS OF Φ(u, v, β)
Now there exists an A > 0 such that (x−m)z e iπ z = λe 4 , (7.12) ez − e−iη < A,
< λ
1 16 .
I2r
= O η σ1 −1 (η + 2πβ)σ2 −1
Hence √ 2cη
2
tλ − 16η 2
e
dλ .
Splitting the integrand as 2 t − 16η 2λ
e
2 t − 32η 2λ
=e
2 t − 32η 2λ
×e
and noting that √
2cη
2 t − 32η 2λ
e
dλ ≤
∞
2 t − 32η 2λ
e
dλ =
0
we ﬁnd I2r
√ η 8π √ , t
2 t σ1 −1 σ2 −1 − 32η2 η √ =O η (η + 2πβ) e . t
This proves (7.11). We now consider I2 . In view of the assumption dist(η, 2πZ) > 8 , we have (x−m)w e (7.16) z < 2 . ez − e−iη = O(1), In particular, ϕ(z) is analytic for z < 2 . Thus we can write ϕ(z) =
∞ n=0
bn z n =
N −1 n=0
where (7.17)
z < 2 ,
bn z n + sN (z),
sN (z) =
zN 2πi
Γ
ϕ(w)dw wN (w − z)
and Γ is a counterclockwise contour which encircles 0 and z but none of the poles of ϕ. We claim that N −1 it 1 ( 1 − )(w−iη)2 bn (w − iη)n dw I2 = e−(m+1)iη (iη)u−1 (iη + 2πiβ)v−1 e 2 η2 (η+2πβ)2 C2
n=0
(7.18)
⎧ N η ⎪ σ1 −1 σ2 −1 22N 6 √ ⎪ (η + 2πβ) , ⎨O η t t + − N2+1 ⎪ ⎪ , ⎩O η σ1 −1 (η + 2πβ)σ2 −1 4N3tη2
1≤N < 1
t3 < η
t1+ . We recall that u = σ1 + it and v = σ2 − it. ) Theorem 8.1 (The asymptotics to all orders of the integral Hˆ α wu−1 × (w + 2iπβ)v−1 e−nw dw). For every > 0, there exists a constant A > 0 such that iπ wu−1 (w + 2iπβ)v−1 e−nw dw = −(iη)u−1 (iη + 2πiβ)v−1 e−inη e 4 ˆα H
−k− 12 φ(2k) (0) t 1 1 1 k i × − Γ k+ (2k)! 2 η2 (η + 2πβ)2 2 k=0 N 7N 6 1 √ , + O η σ1 (η + 2πβ)σ2 −1 t → ∞, t t [ N 2−1 ]
(8.1) t1+ < β < ∞,
n = 1, 2, . . . , [t],
σ1 ∈ [0, 1],
σ2 ∈ [0, 1],
1 ≤ N < At,
ˆ α denotes the Hankel contour deﬁned in ( 7.3), η and φ(z) are given by where H ( 2t η = βπ − 1 + 1 + (8.2) , βπn (8.3)
2 z z 1 1 (u−1) ln(1+ iη )+(v−1) ln(1+ iη+2πiβ )−nz− it 2 ( η 2 − (η+2πβ)2 )z
φ(z) = e
,
and the error terms are uniform with respect to n, σ1 , σ2 , β, N in the given ranges. Proof. Let > 0 be given and suppose that β > t1+ , n = 1, . . . , [t], σ1 ∈ [0, 1], σ2 ∈ [0, 1], and N ≥ 1. All error terms of the form O(·) will be uniform with respect to n, σ1 , σ2 , β, N (but not with respect to ). Let η be given by (8.2). The function x( 1 + 1/x − 1) increases from 0 to 1/2 as x goes from 0 to ∞. Hence, using that ( 2t βπn 2t η= 1+ −1 n 2t βπn we infer that, given any δ > 0, we have t t (8.4) (1 − δ) < η < n n for all suﬃciently large t. 73
74
8. MORE EXPLICIT ASYMPTOTICS OF Φ(u, v, β)
ˆ α into the As in the proof of Theorem 7.1, we deform the Hankel contour H straight lines Cj , j = 1, . . . , 4 joining ∞, iη + (1 + i)cη, iη − (1 + i)cη, −cη − iη, ∞, where 0 < c ≤ 1/2 is an absolute constant. This implies
wu−1 (w + 2iπβ)v−1 e−nw dw =
ˆα H
where
4
Ij ,
j=1
wu−1 (w + 2iπβ)v−1 e−nw dw.
Ij = Cj
We ﬁrst prove that I1 , I3 , I4 are exponentially small as t → ∞. We let Q = > Ant → ∞ as t → ∞. 2πβ/η. Then, by (8.4), Q > 2πβn t Using (7.7) and (8.4), we ﬁnd that I4 is exponentially small: ∞ Q−1 1 I4 = wu−1 (w + 2iπβ)v−1 e−nw dw = O e−t(arctan c +arctan c ) e−nw1 dw1 −cη
C4 −t(arctan
Q−1 1 c +arctan c )+ncη
−t(arctan
1 c −c)
= O(e = O(e
−At
) = O(e
−t(arctan
) = O(e
Q−1 1 c +arctan c −c)
)
). ˜
We next consider I3 . Letting w = ρeiφ and w + 2iπβ = ρ˜eiφ , equations (7.8) and (8.4) yield wu−1 (w + 2iπβ)v−1 e−nw = O((cη)σ1 −1 (cη)σ2 −1 e−t(arctan( = O(e−t(arctan( where
F (Q) = arctan
1−c+Q )−arctan 1−c c c −c)
1−c+Q c
− arctan
1−c+Q )−arctan 1−c c c )
encη )
) = O(e−tF (Q) ),
1−c c
− c.
Since Q → ∞ and F (Q) > A > 0 for all suﬃciently large Q, it follows that I3 = O(e−At ). For w ∈ C1 we have w = w1 + i(1 + c)η with w1 ≥ cη. Equations (7.9) and (8.4) imply ˜ wu−1 (w + 2iπβ)v−1 e−nw = O η σ1 −1 (η + 2πβ)σ2 −1 e−t(φ−φ)−nw1 ˜ nw1 ˜ = O e−t(φ−φ+ t ) = O e−t(φ−φ+(1−δ)P ) = O e−tF (P,Q) where P = w1 /η ≥ c and F (P, Q) = arctan
1+c+Q 1+c − arctan + (1 − δ)P. P P
Now the rhs of the inequality F (P, Q) − AP > arctan
1+c π − + (1 − δ)P − AP P 2
8. MORE EXPLICIT ASYMPTOTICS OF Φ(u, v, β)
75
π is an increasing function of P ≥ c and arctan 1+c c − 2 + (1 − δ)c − Ac > 0 for δ and A small enough. Hence F (P, Q) > AP . It follows that I1 = wu−1 (w + 2iπβ)v−1 e−nw dw C1 ∞ ∞ −tAP −tAP e dw1 = O η e dP = O ηt−1 e−cAt = O e−cAt . =O cη
c
1 It remains to analyze the integral I2 . Using that n = t( η1 − η+2πβ ), we write it 1 1 1 ( 1 − )(w−iη)2 e 2 η2 (η+2πβ)2 φ(w − iη)dw, I2 = (iη)u−1 (iη + 2πiβ)v−1 e−itη( η − η+2πβ ) C2
where φ(z) is deﬁned in (8.3). Deﬁne {aj }∞ 0 by φ(z) =
∞
z < η.
aj z j ,
j=0
Then φ(z) =
N −1
aj z j + rN (z),
z < η,
j=0
where
φ(w) zN dw, N 2πi Γ w (w − z) and Γ is a counterclockwise contour contained in the disk of radius η centered at the origin which √ encircles the points 0 and z once. 21 Let α ∈ ( 2c, 1) be a constant. Let z < 20α 21 η (so that 20 z < αη) and let Γ be a circle with center w = 0 and radius ρN , where 21 z ≤ ρN ≤ αη. 20 By (7.13), 2tρ3 2tρ3 N N 2πρN zN N −N 3(1−α)η3 3 3(1−α)η e rN (z) = O = O z ρN e , z ≤ αη. ρN N (ρN − z) rN (z) =
2tρ3
2et N/3 for ρ = ( (1−α)N )1/3 η; The function ρ−N e 3(1−α)η3 has the minimum ( (1−α)N η3 ) 2t ρN can have this value if 1 (1 − α)N 3 21 z ≤ η ≤ αη. 20 2t Hence, (8.5) N3 1 2et 2α3 20 (1 − α)N 3 N η. rN (z) = O z , N ≤ t, z ≤ (1 − α)N η 3 1−α 21 2t
For z
0, there exists a constant A > 0 such that m wu−1 (w + 2iπβ)v−1 e−nw dw = J + O(t1/6 β σ2 −1 ), t → ∞, n=1
ˆα H
m = 1, . . . , [t],
(8.15)
t1+ < β < ∞,
σ1 ∈ [0, 1],
σ2 ∈ [0, 1],
where J is deﬁned by ( (8.16) J = e
iπ 2 (σ1 +σ2 )
e
iπ 4
it m √ √ 2t π (πβ)σ1 +σ2 −1 ( R − 1)σ1 ( R + 1)σ2 F, 2t πnβ n=1
with 2t , R := 1 + πnβ
(8.17)
√ (1 + R)−2it iπnβ(1−√R) F := e , R1/4
and the error term is uniform with respect to m, β, σ1 , σ2 in the given ranges. Proof. Summing equation (8.1) with N = 2 from n = 1 to n = m, we ﬁnd m m 1 σ1 u−1 v−1 −nw σ2 −1 w (w + 2iπβ) e dw = J + O 5/6 η (η + 2πβ) , t ˆ n=1 Hα n=1 where J =−
m
(iη)
n=1
u−1
(iη + 2πiβ)
( 2t η = βπ − 1 + 1 + . βπn
v−1 −inη
e
− 12 √ t 1 1 e − π, 2 2 2 η (η + 2πβ) iπ 4
8. MORE EXPLICIT ASYMPTOTICS OF Φ(u, v, β)
79
Straightforward algebra shows that J can be written as in (8.16). Equation (8.15) follows because, by (8.4), m m σ t 1 σ2 −1 η σ1 (η + 2πβ)σ2 −1 = O β = O tσ1 β σ2 −1 t1−σ1 = O tβ σ2 −1 . n n=1 n=1 Substituting the result of corollary 8.2 into the asymptotic expansion of theorem 7.1, we arrive at the following asymptotic formula for Φ(u, v, β). For brevity of presentation, we state the result only to leading order. Corollary 8.3 (The asymptotics of Φ(u, v, β)). For every > 0, there exists a constant A > 0 such that ( π − iπ (σ1 +σ2 ) iπ 2 4 Φ(u, v, β) = −e e (πβ)σ1 +σ2 −1 2t 1 [t( η1 − η+2πβ )] it √ √ 2t × ( R − 1)σ1 ( R + 1)σ2 F πnβ n=1 − i(2π)
σ1 +σ2 −1 − πi 2 (σ1 +σ2 )
e
η [ 2π ]
nu−1 (n + β)v−1 + O(t1/6 β σ2 −1 ),
t → ∞,
n=1
(8.18) 1 ≤ η < t 3 − < ∞, 1
t1+ < β < ∞,
dist(η, 2πZ) > ,
σ1 ∈ [0, 1],
σ2 ∈ [0, 1],
where R and F are deﬁned in ( 8.17) and the error term is uniform with respect to η, β, σ1 , σ2 in the given ranges. Proof. Letting N = 2 in Theorem 7.1 and using Corollary 8.2 we ﬁnd equation (8.18) but with the error term 1 η O(t1/6 β σ2 −1 ) + η σ1 −1 (η + 2πβ)σ2 −1 O t− 3 √ t − 12 iπ √ 1 1 (iη)u−1 (iη + 2πiβ)v−1 e 4 t 1 1 + e− [t( η − η+2πβ )]+1 iη − π. −iη 2 2 1−e 2 η (η + 2πβ) Since this error term is σ1 σ2 −1 − 5 1/6 σ2 −1 σ1 −1 σ2 −1 η 6 √ = O t1/6 β σ2 −1 +O η β +O η t β O t β t the result follows.
CHAPTER 9
Fourier coeﬃcients of the product of two Hurwitz zeta functions The erratic behavior of the Riemann zeta function in the critical strip makes it diﬃcult to obtain good asymptotic information on ζ(s) for 0 ≤ Re s ≤ 1. Much of the literature on the asymptotics of ζ(s) therefore instead focuses on the asymptotics of various related quantities in which the erratic behavior has been smoothed out by averaging. The prime example is the study of the large T behavior of the moments (see [26, chapter VII] and e.g. [7, 25]) T ζ(σ + it)2k dt. 1
A second example involves the small δ behavior of the integral ∞ ζ(σ + it)2k e−δt dt, 0
where δ > 0, see [26, Section 7.12]. Another class of examples, which is relevant for the present chapter, considers the large t behavior of various quantities involving the Hurwitz zeta function ζ(s, α) averaged with respect to the extra parameter α. Recall that the Hurwitz zeta function ζ(s, α) is deﬁned as the analytic continuation of the sum ∞ 1 , Re s > 1, Re α > 0. ζ(s, α) = (n + α)s n=0 For each α with Re α > 0, ζ(s, α) is a meromorphic function of s ∈ C with a simple pole at s = 1 and no other poles. For each s ∈ C \ {1}, ζ(s, α) is analytic in the halfplane Re α > 0. These properties of the Hurwitz function follow easily from the representation eαz z s−1 Γ(1 − s) (9.1) dz, s ∈ C \ {1}, Re α > 0, ζ(s, α) = z 2πi H1 1 − e where H1 is the Hankel contour surrounding the negative real axis deﬁned in (1.13). For many purposes it is more convenient to work with the modiﬁed Hurwitz function ζ1 (s, α) deﬁned by (9.2)
ζ1 (s, α) = ζ(s, α) − α−s = ζ(s, α + 1),
which is regular at α = 0 (in fact, analytic for Re α > −1). For each s ∈ C \ {1}, ζ1 (s, α) is a smooth function of α ∈ [0, 1]. Thus, for each choice of u, v ∈ C \ {1}, the product ζ1 (u, α)ζ1 (v, α) can be represented by its Fourier series ζ1 (u, α)ζ1 (v, α) = (9.3) qn (u, v)e2πinα , 0 < α < 1, n∈Z 81
82
9. FOURIER COEFFICIENTS OF THE PRODUCT
where the Fourier coeﬃcients qn (u, v) are deﬁned by 1 ζ1 (u, α)ζ1 (v, α)e−2πinα dα, qn (u, v) =
n ∈ Z.
0
By standard Fourier analysis, the series in (9.3) converges in L2 ([0, 1]) and converges pointwise for each α ∈ (0, 1). In this chapter, we will show the following two theorems on the Fourier coeﬃcients qn (u, v). The proofs are presented in sections 9.3 and 9.4, respectively. Theorem 9.1 (Expression for qn (u, v)). For each n ∈ Z, the nth Fourier coefﬁcient qn (u, v) can be expressed as follows for u, v ∈ C \ {1} with Re u, Re v < 2: (9.4)
qn (u, v) = bn (u + v) + Rn (u, v) + Rn (v, u) − Tn (u, v) − Tn (v, u),
where, for any n ∈ Z, bn (s), Rn (u, v), and Tn (u, v) are the meromorphic continuations of the functions ∞ (9.5a) bn (s) = α−s e−2πinα dα, Re s > 1, 1 ∞ (9.5b) Rn (u, v) = α−v ζ1 (u, α)e−2iπnα dα, Re(u + v) > 2, Re v < 1, (9.5c)
0 1
Tn (u, v) =
α−v ζ1 (u, α)e−2πinα dα,
Re v < 1.
0
These meromorphic continuations are given by Γ(1 − s, 2πin)(2πin)s−1, n = 0, bn (s) = (9.6a) 1 n = 0, s−1 , ⎧ πiv ie Φ(u,v,n) ⎪ ⎪ 2 sin(πu) , Γ(1 − v) ⎨ ie−πiv Φ(u,v,n) (9.6b) Rn (u, v) = × , 2 sin(πu) ⎪ Γ(u) ⎪ ⎩Γ(u + v − 1)ζ(u + v − 1),
(9.6c)
Tn (u, v) =
n ≥ 1, n ≤ −1, n = 0,
u ζ(u) − 1 + α1−v ζ1 (u + 1, α)e−2iπnα dα 1−v 1−v 0 2iπn 1 1−v α ζ1 (u, α)e−2iπnα dα, Re v < 2, + 1−v 0 1
n ∈ Z,
where Φ(u, v, n) is the function deﬁned in ( 1.12) and Γ(s, z) denotes the incomplete Gamma function: ∞ ∞ Γ(s, z) = r s−1 e−r dr, Γ(s, 0) = Γ(s) = r s−1 e−r dr. z
0
Theorem 9.2 (Large t asymptotics of qn (u, v)). Let u = σ1 +it and v = σ2 −it. For each integer n ≥ 1, the nth Fourier coeﬃcient qn (u, v) satisﬁes the following asymptotic formula as t → ∞: ζ(u) ζ(v) Γ(1 − v) ieπiv Φ(u, v, n) − − Γ(u) 2 sin(πu) 1−v 1−u ζ(u + 1)u ζ(v + 1)v − − + O(t−1 ), σ1 , σ2 ∈ [0, 1], (1 − v)(2 − v) (1 − u)(2 − u)
(9.7) qn (u, v) = bn (u + v) +
where the error term is uniform for σ1 , σ2 in the given range.
9.1. COROLLARIES
83
For n = 0, the following formulas hold as t → ∞: • For σ1 , σ2 ∈ [0, 1] with σ1 + σ2 = 1,
Γ(1 − u) Γ(1 − v) 1 q0 (u, v) = + Γ(σ1 + σ2 − 1)ζ(σ1 + σ2 − 1) + σ1 + σ2 − 1 Γ(v) Γ(u) ζ(v) ζ(u + 1)u ζ(v + 1)v ζ(u) − − − + O(t−1 ), (9.8) − 1 − v 1 − u (1 − v)(2 − v) (1 − u)(2 − u) where the error term is uniform for all σ1 , σ2 ∈ [0, 1] such that σ1 +σ2 = 1. • For σ1 , σ2 ∈ [0, 1] with σ1 + σ2 = 1, ζ(u) ζ(v) t +γ− − 2π 1−v 1−u ζ(u + 1)u ζ(v + 1)v − − + O(t−1 ), (1 − v)(2 − v) (1 − u)(2 − u)
(9.9)
q0 (u, v) = ln
where the error term is uniform for all σ1 , σ2 ∈ [0, 1] such that σ1 +σ2 = 1, and γ denotes the Euler constant. Remark 9.3. The results of theorem 9.1 and 9.2 are wellknown in the case when n = 0. More precisely, the formulas (9.9) and (9.8) can be found in Theorem 1 and Theorem 2 of [29], respectively. They are included here for the sake of comparison; the new content lies in the case of nonzero n. Note however that the ζ(u+1)u ζ(v+1)v and (1−u)(2−u) in (9.8) and (9.9) are missing in [29]. These terms terms (1−v)(2−v) must be included in order to make the asymptotic formulas uniform for σ1 and/or σ2 near 0, because it is known that ζ(1 + it) is unbounded as t → ∞; in fact, ζ(1 + it) = Ω(ln ln t), see Titchmarsh [26, Theorem 8.5]. Remark 9.4. The integral in (9.5b) converges whenever Re(u + v) > 2 and Re v < 1, because for each s ∈ C \ {1}, ζ1 (s, α) satisﬁes the large α asymptotics α−s α1−s − + O(α−s−1 ), s−1 2 uniformly as α → ∞ in the halfplane Re α > 0, see e.g. [19, Eq. (25.11.43)]. (9.10)
ζ1 (s, α) =
9.1. Corollaries By substituting the asymptotic formula for Φ(u, v, n) found in corollary 7.2 into (9.7), we obtain the following corollary of theorem 9.2. Corollary 9.5. Let u = σ1 + it and v = σ2 − it. For each integer n ≥ 1, Γ(1 − v) ieπiv Γ(u) 2 sin(πu)
m wu−1 (w + 2iπn)v−1 e−jw dw × − e−πi(σ1 +σ2 )
qn (u, v) = bn (σ1 + σ2 ) + (9.11)
j=1 πi 4
ˆα H
− πi 2 (σ1 +σ2 )
(−1)m e e
π σ1 +σ2 − 2 (1 + 2n)v 1
−3/2
+ O(t ) 2 2n(1 + n)t ζ(u) ζ(v) ζ(u + 1)u ζ(v + 1)v − − − − + O(t−1 ), 1−v 1 − u (1 − v)(2 − v) (1 − u)(2 − u) σ1 , σ2 ∈ [0, 1], t → ∞,
+
84
9. FOURIER COEFFICIENTS OF THE PRODUCT
t where m = [ π(1+(2n) −1 ) ] and the error term is uniform for σ1 , σ2 ∈ [0, 1].
In the special case of σ1 = σ2 , we obtain asymptotic estimates for the Fourier coeﬃcients 1 ζ1 (u, α)2 e−2πinα dα Qn (u) = 0
of ζ1 (s, α)2 as corollaries. Corollary 9.6 (Large t asymptotics of Q0 (u)). For σ ∈ [0, 1] with σ = 1/2, 1 + 2Γ(2σ − 1)ζ(2σ − 1) sin(πσ)t1−2σ 2σ − 1 Im ζ(σ + 1 + it) ζ(σ + it) −2 + O(t−1 ), (9.12) − 2 Re t → ∞, 1 − σ + it t where the error term is uniform with respect to σ in the given range. For σ = 1/2, ζ( 1 + it) 1 t Q0 + it = ln + γ − 2 Re 12 (9.13) t → ∞. + O(t−1 ), 2 2π + it 2 Q0 (σ + it) =
Proof. Let u = σ + it and v = σ − it. Then (see (A.1)) Γ(1 − u) Γ(1 − v) + = 2 sin(πσ)t1−2σ + O(t−2σ−1 ) Γ(v) Γ(u) uniformly for σ ∈ [0, 1]. Hence (9.12) and (9.13) follow from equations (9.8) and (9.9), respectively. Corollary 9.7. As t → ∞, (9.14)
⎧ 1−2σ ⎪ ), ⎨O(t Q0 (σ + it) = O(ln t), ⎪ ⎩ O(1),
σ ∈ [0, 1/2), σ = 1/2, σ ∈ (1/2, 1].
If the error terms O(t1−2σ ) and O(1) on the rhs are replaced with O(t1−2σ ln t) and O(ln t), respectively, then ( 9.14) holds uniformly for σ ∈ [0, 1]. Proof. Equation (9.14) follows immediately from corollary 9.6. From this corollary it also follows that the error terms in (9.14) are uniform with respect to σ ∈ [0, 12 − δ] ∪ { 12 } ∪ [ 12 + δ, 1] for any ﬁxed δ > 0. We will complete the proof by showing that (9.15)
Q0 (σ + it) = O(t1−2σ ln t),
t → ∞,
uniformly for σ ∈ [0, 1/2) and (9.16)
Q0 (σ + it) = O(ln t),
t → ∞,
uniformly for σ ∈ (1/2, 1]. By (9.12), we have 1 + 2Γ(2σ − 1)ζ(2σ − 1) sin(πσ)t1−2σ + O(1), t → ∞, 2σ − 1 uniformly for σ ∈ [0, 1/2) ∪ (1/2, 1]. Since σ 1 + O((1 − 2σ)t1−2σ ln t), σ ∈ [0, 1/2), 1−2σ 1−2σ t =1−2 t (ln t)dσ = 1 + O((2σ − 1) ln t), σ ∈ (1/2, 1], 1/2 Q0 (σ + it) =
9.1. COROLLARIES
85
uniformly for σ in the given ranges, and since 1 + 2Γ(2σ − 1)ζ(2σ − 1) sin(πσ), 2σ − 1
2Γ(2σ − 1)ζ(2σ − 1) sin(πσ)2σ − 1,
are bounded functions of σ ∈ [0, 1], we ﬁnd (9.15) and (9.16).
Corollary 9.8 (Large t asymptotics of Qn (u)). Let u = σ + it. For each integer n ≥ 1, Qn (u) = Q−n (u) = bn (2σ) − t1−2σ ie−πiσ
m j=1
(9.17)
+ t 2 −2σ 1
m
πi 4
wu−1 (w + 2iπn)u¯−1 e−jw dw
ˆα H
2σ− 12
Im ζ(σ + 1 + it) i(−1) e π (1 + 2n)u¯ ζ(σ + it) −2 − 2 Re 1 − σ + it t 2 2n(1 + n)
+ O(t−1 ) + O(t− 2 −2σ ), 1
t → ∞,
σ ∈ [0, 1],
t where m = [ π(1+(2n) −1 ) ] and the error term is uniform for σ in the given range.
Proof. Let u = σ + it and v = σ − it. By (A.1), the coeﬃcient of Φ(u, v, n) in (9.7) satisﬁes πi Γ(1 − v) ieπiv = t1−2σ e 2 (2σ+1) 1 + O(t−2 ) , Γ(u) 2 sin(πu)
t → ∞,
uniformly for σ ∈ [0, 1]. Hence there exist constants T ≥ 1 and c > 0 such that Γ(1 − v) ieπiv 1−2σ , Γ(u) 2 sin(πu) ≥ ct for all t ≥ T and all σ ∈ [0, 1]. Since the terms not involving Φ on the rhs of (9.7) are O(1), and clearly Qn (u) ≤ Q0 (u) for each integer n, we conclude from (9.7) and corollary 9.7 that O(t1−2σ ln t), σ ∈ [0, 1/2), 1−2σ t Φ(u, v, n) = O(Q0 (u)) + O(1) = (9.18) O(ln t), σ ∈ [1/2, 1]. uniformly for σ ∈ [0, 1]. Equation (9.17) now follows from (9.11).
From (9.18) and corollary 9.7, we also obtain an estimate for Φ. Corollary 9.9. For each integer n ≥ 1, ⎧ ⎪ ⎨O(1), (9.19) Φ(σ + it, σ − it, n) = O(ln t), ⎪ ⎩ O(t2σ−1 ),
σ ∈ [0, 1/2), σ = 1/2, σ ∈ (1/2, 1].
If the error terms O(1) and O(t2σ−1 ) on the rhs are replaced with O(ln t) and O(t2σ−1 ln t), respectively, then ( 9.19) holds uniformly for σ ∈ [0, 1].
86
9. FOURIER COEFFICIENTS OF THE PRODUCT
9.2. Asymptotics of the zeroth Fourier coeﬃcient In order to put theorems 9.1 and 9.2 in context, let us ﬁrst consider the case n = 0 of the zeroth Fourier coeﬃcient. The study of the large t asymptotics of the zeroth Fourier coeﬃcient 1 Q0 (s) = ζ1 (s, α)2 dα, s = σ + it, 0
i.e., of the mean square of ζ1 (s, α) as a function of α ∈ [0, 1], has a long history. Koksma and Lekkerkerker showed in the 1952 paper [17] that Q0 ( 12 + it) = O(ln t) and that 1 1 1 1−2σ +O t + ln t uniformly for < σ ≤ 1. Q0 (σ + it) = 2σ − 1 2σ − 1 2 In the case when σ = 1/2, Balasubramanian established [4] the asymptotic formula 1 Q0 + it = ln t + O(ln ln t) 2 and in the 1980s the error term in this formula was improved to O(1) by Rane [21]. Sitaramachandrarao was awarded a prize (announced in HardyRamanujan Journal 10 (1987), p. 28) for proving the more precise result 1 t Q0 + it = ln + γ + O(t−3/16 (ln t)3/8 ). 2 2π Zhang independently obtained [27] a similar result but with the slightly larger error term O(t−3/16 (ln t)11/8 ), which was later improved to O(t−7/36 (ln t)25/18 ) in [28]. Finally, in [1] and [29], Andersson and Zhang independently arrived at the asymptotic expansion stated in (9.13). Around the same time as Andersson and Zhang obtained the expansion (9.13), Katsurada and Matsumoto started using Atkinson’s dissection argument applied to the product ζ1 (u, α)ζ1 (v, α) with two independent complex variables u and v in order to analyze Q0 , see [14]. In [15], they reﬁned their approach and found an alternative proof of (9.13) together with a number of other related results. The derivation of [15] is based on the following expression for the zeroth Fourier coeﬃcient (see [15, Eq. (2.1)]): (9.20)
q0 (u, v) =
1 + R0 (u, v) + R0 (v, u) − T0 (u, v) − T0 (v, u), u+v−1 − N + 1 < Re u, Re v < N + 1,
where (9.21)
R0 (u, v) = Γ(u + v − 1)ζ(u + v − 1)
Γ(1 − v) Γ(u)
and, for any integer N ≥ 0, T0 (u, v) = (9.22)
N −1
(u)k (ζ(u + k) − 1) (1 − v)k+1 k=0 ∞ (u)N 1−u−v ∞ u+v−2 + l β (1 + β)−u−N dβ, (1 − v)N l l=1
9.2. ASYMPTOTICS OF THE ZEROTH FOURIER COEFFICIENT
87
with the Pochhammer symbol (a)k being deﬁned by (a)k =
Γ(a + k) = a(a + 1)(a + 2) · · · (a + k − 1). Γ(a)
It is not hard to see that T0 is independent of the choice of N (see Lemma 9.10). It appears that the identity (9.20) provides the most powerful approach available for determining the asymptotics of Q0 (s). For example, the expansion of corollary 9.6 for Q0 (σ + it) with σ = 1/2 is easily obtained from (9.20) by taking u = v¯ = σ + it and N = 1, and noting that the sum over l in (9.22) is O(t−1 ); the expansion for σ = 1/2 is obtained in a similar way by taking N = 1 and σ = (1 + δ)/2 and letting δ → 0, see [15] or the proof of theorem 9.2. The expression (9.4) for qn given in theorem 9.1 is the natural generalization to 1 clearly any integer n ∈ Z of the expression (9.20) for q0 . Indeed, since b0 (s) = s−1 ) ∞ −s is the analytic continuation of the integral 1 α dα, this follows from (9.5) and the following lemma. Lemma 9.10. The functions R0 and T0 deﬁned in ( 9.21) and ( 9.22) are the analytic continuations of the integrals ∞ α−v ζ1 (u, α)dα, Re(u + v) > 2, Re v < 1, R0 (u, v) =
0 1
T0 (u, v) =
α−v ζ1 (u, α)dα,
Re v < 1.
0
Proof. Employing the integral representation ∞ −αr u−1 e r 1 (9.23) dr, Re u > 1, ζ1 (u, α) = Γ(u) 0 er − 1
Re α > −1,
and using Fubini’s theorem to change the order of integration, we ﬁnd, for Re(u + v) > 2 and Re v < 1, ∞ ∞ u−1 ∞ r 1 −v α ζ1 (u, α)dα = α−v e−αr dαdr. r −1 Γ(u) e 0 0 0 The change of variables β = αr shows that the rhs can be rewritten as ∞ u+v−2 ∞ r 1 β −v e−β dβdr. Γ(u) 0 er − 1 0 Since the βintegral equals Γ(1 − v), we arrive at ∞ Γ(1 − v) ∞ r u+v−2 −v dr α ζ1 (u, α)dα = Γ(u) er − 1 0 0 Γ(1 − v) = Γ(u + v − 1)ζ(u + v − 1), Re(u + v) > 2, Γ(u)
Re v < 1,
which proves the statement for R0 . Integrating by parts repeatedly using the facts that ζ1 (s, 1) = ζ(s) − 1 and (9.24)
∂α ζ1 (s, α) = −sζ1 (s + 1, α),
88
9. FOURIER COEFFICIENTS OF THE PRODUCT
we obtain, for any integer N ≥ 1 and Re v < 1, 1 1 u ζ(u) − 1 + α−v ζ1 (u, α)dα = α1−v ζ1 (u + 1, α)dα 1 − v 1 − v 0 0 1 N −1 (u)k (ζ(u + k) − 1) (u)N = ··· = (9.25) + αN −v ζ1 (u + N, α)dα. (1 − v)k+1 (1 − v)N 0 k=0
The statement for T0 will follow once we prove that (9.26)
1
α
N −v
ζ1 (u + N, α)dα =
∞
0
l=1
l
1−u−v
∞
β u+v−2 (1 + β)−u−N dβ
l
for Re v < N + 1 and Re u > −N + 1. But for Re u > −N + 1 we can use the representation (9.27)
ζ1 (s, α) =
∞ l=1
1 , (l + α)s
Re s > 1,
Re α > −1,
to write the lefthand side of (9.26) as ∞
1
αN −v (l + α)−u−N dα.
0
l=1
The change of variables α = l/β completes the proof of (9.26) and hence also of the lemma. Remark 9.11. The formula for Tn given in (9.6c) is obtained by integrating by parts once in the integral (9.5c). If we instead integrate by parts N times, we ﬁnd 1 N −1 (−1)k ∂ k α−v ζ1 (u, α)e−2πinα dα = (ζ1 (u, α)e−2πinα ) Tn (u, v) = k (1 − v) ∂α k+1 0 α=1 k=0 N 1 N ∂ (−1) + (9.28) αN −v N (ζ1 (u, α)e−2πinα )dα (1 − v)N 0 ∂α k N −1 k (u)j (2πin)k−j (ζ(u + j) − 1) = j (1 − v)k+1 j=0 k=0
+
N N (u)j (2πin)N −j j=0
j
(1 − v)N
1
αN −v ζ1 (u + j, α)e−2πinα dα.
0
Using that (cf. (9.26)), for 2 ≤ j ≤ N with Re v < N + 1 and Re u > −j + 1, 1 αN −v ζ1 (u + j, α)e−2πinα dα 0
=
∞ l=1
l1−u−v+N −j
l
∞
β u+v−2−N +j (1 + β)−u−j e−2πinl/β dβ,
9.3. PROOF OF THEOREM 9.1
89
this leads to the following expression for Tn when Re u > −1 and Re v < N + 1, which is the natural extension to any integer n of the expression (9.22) for T0 : N −1 N k k (u)j (2πin)k−j N (u)j (2πin)N −j (ζ(u + j) − 1) + Tn (u, v) = j j (1 − v)k+1 (1 − v)N j=2 k=0 j=0 ∞ ∞ (9.29) × l1−u−v+N −j β u+v−2−N +j (1 + β)−u−j e−2πinl/β dβ l=1
N
+
(2πin) (1 − v)N
l 1
0 N −1
+
u(2πin) (1 − v)N
αN −v ζ1 (u, α)e−2πinα dα
1
αN −v ζ1 (u + 1, α)e−2πinα dα.
0
9.3. Proof of Theorem 9.1 Lemma 9.12 (Fourier series of ζ1 (u, α)ζ1 (v, α)). Suppose u, v ∈ C \ {1} and δ ∈ (0, π/2). Then the Fourier series representation of ζ1 (u, ·)ζ1 (v, ·) ∈ C ∞ ([0, 1]) is given by ζ1 (u, α)ζ1 (v, α) = (9.30) qn (u, v)e2πinα , n∈Z
where (9.31)
e−(sgn n)iδ ∞
qn (u, v) =
−u−v α + α−v ζ1 (u, α) + α−u ζ1 (v, α) e−2πinα dα,
1
n ∈ Z \ {0}, and
1
q0 (u, v) =
ζ1 (u, α)ζ1 (v, α)dα. 0 2
The series in ( 9.30) converges in L ([0, 1]) and converges pointwise for each α ∈ (0, 1). If Re(u + v) > 2, the Fourier coeﬃcients can be expressed as ∞ −u−v α qn (u, v) = (9.32) + α−v ζ1 (u, α) + α−u ζ1 (v, α) e−2πinα dα, 1
n ∈ Z,
u, v ∈ C \ {1},
Re(u + v) > 2.
Proof. Equation (9.31) follows from (9.2) and easy algebra. Indeed, for u, v ∈ C \ {1} and n ≥ 1, we compute 1 ζ1 (u, α)ζ1 (v, α)e−2πinα dα qn (u, v) = 0
e−iδ ∞
= 0
−
e−iδ ∞
ζ1 (u, α)ζ1 (v, α)e−2πinα dα
1
e−iδ ∞
=
ζ(u, α + 1)ζ(v, α + 1)e−2πinα dα
0
e−iδ ∞
− 1
(ζ(u, α) − α−u )(ζ(v, α) − α−v )e−2πinα dα.
90
9. FOURIER COEFFICIENTS OF THE PRODUCT
Changing variables β = α + 1 in the integral from 0 to e−iδ ∞, we infer that e−iδ ∞ e−iδ ∞ −2πinβ qn (u, v) = ζ(u, β)ζ(v, β)e dβ − ζ(u, α)ζ(v, α)e−2πinα dα 1
1 e−iδ ∞
+ =
−v α ζ(u, α) + α−u ζ(v, α) − α−u−v e−2πinα dα
1 e−iδ ∞
−v α ζ1 (u, α) + α−u ζ1 (v, α) + α−u−v e−2πinα dα,
1
which proves (9.31) for n ≥ 1; the same computation applies also when n ≤ −1 if δ is replaced by −δ. The expression in (9.32) follows because, by (9.10), ζ(s, α) = α1−s −s s−1 + O(α ) as α → ∞, so if Re(u + v) > 3 and u, v = 1, the above proof can be repeated for each n ∈ Z with δ = 0; the resulting formula can then be extended to Re(u + v) > 2 by analytic continuation. Proof of Theorem 9.1. Equations (9.4) and (9.5) are a direct consequence of (9.32). It remains to establish the meromorphic continuations in (9.6). For n = 0, this was carried out already in section 9.2 (see Lemma 9.10 and equation (9.25) with N = 1). Suppose n is a nonzero integer. Making the change of variables r = 2πinα in the integral in (9.5a) and deforming the contour back to the real axis, we obtain, for Re s > 1, ∞ r −s (2πin)s−1 e−r dr = Γ(1 − s, 2πin)(2πin)s−1. bn (s) = 2πin
This proves (9.6a). To determine the analytic continuation of Rn , we substitute the integral representation (9.23) for ζ1 into (9.5b) and change the order of integration. This gives, for Re(u + v) > 2 and Re v < 1, ∞ ∞ u−1 ∞ r 1 α−v ζ1 (u, α)e−2πinα dα = α−v e−α(r+2πin) dαdr. Γ(u) 0 er − 1 0 0 The change of variables β = α(r + 2πin) shows that the rhs can be rewritten as ∞ ∞ u−1 r 1 v−1 (r + 2πin) β −v e−β dβdr, Γ(u) 0 er − 1 0 where we have used the decay of e−β to deform the contour back to the positive real axis. Since the βintegral equals Γ(1 − v), we arrive at the formula ∞ Γ(1 − v) ∞ r u−1 (r + 2πin)v−1 dr, (9.33) α−v ζ1 (u, α)e−2πinα dα = r −1 Γ(u) e 0 0 Re(u + v) > 2, Re v < 1. The integral on the rhs can be expressed in terms of Φ: ∞ u−1 1 , r v−1 e−iπ(u+v) −eiπ(u−v) (r + 2πin) dr = Φ(u, v, n) × (9.34) 1 r −1 e , 0 e−iπ(u−v) −eiπ(u+v)
n > 0, n < 0,
for Re u > 1 and v ∈ C. Indeed, for Re u > 1 the contribution from the circular part of the Hankel contour Hα in (1.12) vanishes in the limit α → 0; hence (1.12)
9.4. PROOF OF THEOREM 9.2
can be written as iπu
Φ(u, v, n) = (e
−iπu
−e
) 0
∞
91
r u−1 (−r − 2πin)v−1 dr. er − 1
For n > 0 (resp. n < 0), −r − 2πin lies in the third (resp. second) quadrant, and so (−r − 2πin)v−1 = e−(sgn n)iπ(v−1) (r + 2πin)v−1 , which proves (9.34). The expression (9.6b) for Rn follows from (9.33) and (9.34). The expression (9.6c) for Tn is the special case N = 1 of (9.28). 9.4. Proof of Theorem 9.2 Lemma 9.13. Let u = σ1 + it and v = σ2 − it. Then, for every N ≥ 1, Rn (v, u) = O(t−N ) uniformly for n ≥ 1 and σ1 , σ2 ∈ [0, 1] as t → ∞. Proof. This is an immediate consequence of the expression (9.6b) for Rn , the estimate (7.30) of Φ(v, u, β), and the fact that (see (A.2)) Γ(1 − u) ieπiu = O(t1−σ1 −σ2 e−2πt ), Γ(v) 2 sin(πv)
t → ∞,
uniformly for σ1 , σ2 ∈ [0, 1].
Lemma 9.14. Let u = σ1 + it, v = σ2 − it, and n ∈ Z. Then, as t → ∞, (9.35a) (9.35b)
ζ(u) − 1 (ζ(u + 1) − 1)u + + O(t−1 ), 1−v (1 − v)(2 − v) ζ(v) − 1 (ζ(v + 1) − 1)v + + O(t−1 ), Tn (v, u) = 1−u (1 − u)(2 − u)
Tn (u, v) =
uniformly for σ1 , σ2 ∈ [0, 1]. Proof. Consider the expression (9.29) for Tn for some integer N ≥ 3. An integration by parts shows that the integral with respect to dβ in (9.29) can be written as follows for Re v < N + 1: ∞ (1 + l)1−u−j u+v−2−N +j l (9.36) β u+v−2−N +j (1 + β)−u−j e−2πinl/β dβ = u+j−1 l ∞ (1 + β)1−u−j ∂ u+v−2−N +j −2πinl/β dβ = E1 + E2 + E3 , β − e 1 − u − j ∂β l where we have used the shorthand notation (1 + l)1−u−j u+v−2−N +j E1 := l , u+j−1 ∞ (1 + β)1−u−j u+v−3−N +j −2πinl/β β E2 := (u + v − 2 − N + j) e dβ, u+j−1 l ∞ (1 + β)1−u−j u+v−4−N +j −2πinl/β β E3 := 2πinl e dβ. u+j−1 l Substituting (9.36) into (9.29), we see that Tn (u, v) can be expressed for Re u > −1 and Re v < N + 1 as N −1 5 k k (u)j (2πin)k−j (9.37) (ζ(u + j) − 1) + Fm , Tn (u, v) = j (1 − v)k+1 m=1 j=0 k=0
92
9. FOURIER COEFFICIENTS OF THE PRODUCT
where we have used the shorthand notation N ∞ N (u)j (2πin)N −j 1−u−v+N −j Fm := l Em , j (1 − v)N j=2 l=1 1 (2πin)N F4 := αN −v ζ1 (u, α)e−2πinα dα, (1 − v)N 0 u(2πin)N −1 1 N −v F5 := α ζ1 (u + 1, α)e−2πinα dα. (1 − v)N 0
m = 1, 2, 3,
The ﬁrst two terms on the rhs of (9.35a) come from the terms with k = j = 0 and k = j = 1 in the double sum in (9.37). All other terms in this double sum are O(t−1 ) by standard estimates of ζ(s). Thus the asymptotic formula (9.35a) for Tn (u, v) will follow if we can show that Fm = O(t−1 ) uniformly for σ1 , σ2 ∈ [0, 1] for each m = 1, . . . , 5. The terms F1 , F2 , and F3 are easily estimated: F1  ≤
N ∞ C −1 l (1 + l)1−σ1 −j = O(t−1 ), t j=2 l=1
F2  ≤ C
N ∞
l1−σ1 −σ2 +N −j
C t
F3  ≤ C
∞ N
l2−σ1 −σ2 +N −j
≤
C t
∞
l
j=2 l=1 ∞ N
(1 + β)1−σ1 −j σ1 +σ2 −3−N +j β dβ u + j − 1
l−σ1 −j = O(t−1 ),
j=2 l=1 N ∞
∞
l
j=2 l=1
≤
(1 + β)1−σ1 −j σ1 +σ2 −4−N +j β dβ u + j − 1
l−σ1 −j = O(t−1 ),
j=2 l=1
uniformly for σ1 , σ2 ∈ [0, 1]. Integrating by parts twice and using (9.24), we ﬁnd, for σ1 , σ2 ∈ [0, 1], 1 (9.38) αN −v e−2πinα ζ1 (u, α)dα = ∂α f (v, 1)ζ1 (u, 1) + uf (v, 1)ζ1 (u + 1, 1) 0
1
f (v, α)ζ1 (u + 2, α)dα,
+ u(u + 1) 0
where f (v, α) is deﬁned by
α
f (v, α) = 0
α1
α2 N −v e−2πinα2 dα2 dα1 .
0
The functions f (v, α) and ∂α f (v, α) are uniformly bounded for α ∈ [0, 1] and Re v ∈ [0, 1]. Also, by a trivial estimate of the sum in (9.27), ζ1 (u + 2, α) is uniformly bounded for α ∈ [0, 1] and Re u ∈ [0, 1]. Since ζ1 (u, 1) = ζ(u) − 1 and ζ1 (u + 1, 1) = ζ(u + 1) − 1, we conclude that the rhs of (9.38) is O(t2 ) uniformly for σ1 , σ2 ∈ [0, 1]. It follows that F4 = O(t2−N ) = O(t−1 ) uniformly for σ1 , σ2 ∈ [0, 1] as t → ∞. A similar argument shows that F5 also is O(t2−N ). This completes the
9.5. FOURIER COEFFICIENTS OF ζ(s, α) AND ζ1 (s, α)
93
proof of the estimate (9.35a) for Tn (u, v); the estimate (9.35b) for Tn (v, u) follows by interchanging u and v in the above arguments. Proof of Theorem 9.2. Employing lemma 9.13 and lemma 9.14 in the expression (9.4) for qn (u, v) with n ≥ 1, we ﬁnd formula (9.7). The formula (9.8) for the asymptotics of q0 (u, v) when σ1 + σ2 = 1 follows immediately by substituting the expression (9.6b) for R0 into equation (9.4) and using Lemma 9.14. To derive the formula (9.9) for the asymptotics of q0 (u, v) when σ1 + σ2 = 1, we ﬁrst set u = σ + δ + it and v = 1 − σ + δ − it in equation (9.4) and take the limit δ → 0; this yields q0 (u, v) = − ln 2π + γ +
ψ(u) + ψ(v) + T0 (u, v) + T0 (v, u) 2
where u = σ + it, v = 1 − σ − it, and ψ(s) = Γ (s)/Γ(s). The formula (9.9) then −1 follows from Lemma 9.14 and the fact that ψ(σ ± it) = ln t ± πi ) uniformly 2 + O(t for σ ∈ [0, 1] as t → ∞. 9.5. Fourier coeﬃcients of ζ(s, α) and ζ1 (s, α) In this section, we consider, for completeness, the Fourier coeﬃcients of ζ(s, α) and ζ1 (s, α). In particular, we will see that the coeﬃcients bn (s) deﬁned in (9.5a) are the Fourier coeﬃcients of ζ1 (s, α), cf. [2, 22]. We ﬁrst compute the Fourier coeﬃcients an (s) of ζ(s, α) deﬁned by 1 an (s) = ζ(s, α)e−2πinα dα, n ∈ Z. 0
The coeﬃcients an (s) are welldeﬁned whenever Re s < 1, because the relation ζ(s, α) = ζ(s, α + 1) + α−s implies that ζ(s, α) ∼ α−s as α ↓ 0; hence ζ(s, ·) ∈ L1 ([0, 1]) iﬀ Re s < 1. For 0 < Re s < 1, the next lemma was proved in [22] using diﬀerent techniques. Lemma 9.15 (Fourier series of ζ(s, α)). Suppose Re s < 1. Then the Fourier series representation of ζ(s, ·) ∈ L1 ([0, 1]) ∩ C ∞ ((0, 1]) is given by Γ(1 − s)(2πin)s−1 , n ∈ Z \ {0}, 2πinα an (s)e , an (s) = (9.39) ζ(s, α) = 0, n = 0. n∈Z Proof. For Re s < 0, a computation using the representation (9.1) for ζ(s, α) gives 1 eαz z s−1 Γ(1 − s) 1 −2πinα −2πinα an (s) = ζ(s, α)e dα = e dzdα z 2πi H1 1 − e 0 0 1 Γ(1 − s) z s−1 = eα(z−2πin) dα dz 2πi 1 − ez H1 0 ez−2πin − 1 z s−1 z s−1 Γ(1 − s) Γ(1 − s) dz = dz = − z 2πi 2πi H1 z − 2πin 1 − e H1 z − 2πin Γ(1 − s)(2πin)s−1 , n ∈ Z \ {0}, = (9.40) Re s < 0, 0, n = 0,
94
9. FOURIER COEFFICIENTS OF THE PRODUCT
where, in the last step, we have deformed the Hankel contour H1 to inﬁnity and used the residue theorem. By writing 1 1 an (s) = ζ(s, α + 1)e−2πinα dα + α−s e−2πinα dα, Re s < 0, n ∈ Z, 0
0
we see that each Fourier coeﬃcient an (s) is analytic for Re s < 1. Since the expression in (9.39) for an (s) also is analytic for Re s < 1 for each n, the lemma follows from (9.40) and analytic continuation. Remark 9.16. Since ζ(s, ·) ∈ L1 ([0, 1])∩C ∞ ((0, 1]), the Fourier series in (9.39) converges pointwise for each α ∈ (0, 1). Easy estimates show that the convergence is absolute and uniform for s in compact subsets of Re s < 0 and α ∈ [0, 1]. This reﬂects the fact that ζ(s, α) can be extended continuously to a function ζ(s, ·) ∈ C([0, 1]) with ζ(s, 0) = ζ(s, 1) if Re s < 0 (see (9.2)). Lemma 9.17 (Fourier series of ζ1 (s, α)). Suppose s ∈ C \ {1}. Then the Fourier series representation of ζ1 (s, ·) ∈ C ∞ ([0, 1]) is given by (9.41) bn (s)e2πinα , ζ1 (s, α) = n∈Z
where bn (s), n ∈ Z, are the coeﬃcients in ( 9.6a). Proof. We need to show that 1 Γ(1 − s, 2πin)(2πin)s−1, n = 0, −2πinα ζ1 (s, α)e dα = (9.42) 1 n = 0, 0 s−1 ,
s ∈ C \ {1}.
Actually, since both sides of (9.42) are analytic for s ∈ C \ {1} for each n ∈ Z, it is enough to show (9.42) for Re s < 1. By the deﬁnition (9.2) of ζ1 , we have 1 1 −2πinα ζ1 (s, α)e dα = an (s) − α−s e−2πinα dα, Re s < 1. 0
0
The change of variables r = 2πinα shows that 1 − α−s e−2πinα dα = (Γ(1 − s, 2πin) − Γ(1 − s))(2πin)s−1 ,
Re s < 1.
0
Recalling the expression for an (s) in (9.39), the lemma follows.
Remark 9.18. Since ζ1 (s, ·) ∈ C ∞ ([0, 1]), the Fourier series in (9.41) converges pointwise for each α ∈ (0, 1). For ﬁxed s, the incomplete Gamma function admits the asymptotic expansion (see e.g. [19, Eq. (8.11.2)]) N (s − 1) · · · (s − k) s−1 −z −N −1 + O(z ) , z → ∞, Γ(s, z) = z e 1+ zk k=1
where the error term is uniform for  arg z ≤ bn (s) = (2πin)
−1
(1 + O(n
3π 2 −1
− δ. Hence )),
n → ∞,
showing that the Fourier series for ζ1 is not absolutely convergent for any s ∈ C\{1}.
Part 3
Representations for the Basic Sum
CHAPTER 10
Several Representations for the Basic Sum The purpose of this chapter is to present integral representations of the basic b s−1 for certain values of a and b. an Let the contour Ctη , t < η, denote the semicircle from it to iη with Re z ≥ 0. Splitting the contour of the second integral in the rhs of equation (1.2) into the contour Cηt plus the ray from it to ∞ exp(iφ2 ), we ﬁnd
sum
η [ 2π ]
ζ(1 − s) =
n=1
ηs e− 2 − − s(2π)s (2π)s iπs
n
s−1
+ GL (t, σ; η) + GU (t, σ; t),
(10.1)
Ctη
z s−1 dz ez − 1 0 ≤ σ ≤ 1,
0 < t < η,
where GL and GU are deﬁned in (3.5) and (3.6) respectively. The term GU (t, σ; t) was computed to all orders as t → ∞ in theorem 3.2, see equations (3.26) and (3.33). The term GL (t, σ; η) with t < η was computed to all orders as t → ∞ in (3.13). Thus, by comparing equation (10.1) with the representation obtained in [η/2π] theorem 3.2, it follows that we can express the fundamental sum n=[t/2π]+1 ns−1 in terms of the integral appearing in the rhs of (10.1), where the error is computed to all orders. For brevity of presentation, we state this result only to leading order. Lemma 10.1 (An integral representation for the basic sum). Let Ctη , t < η, denote the semicircle from z = it to z = iη with Re z ≥ 0. For every > 0, η [ 2π ]
(10.2)
e− 2 = (2π)s iπs
n
s−1
t n=[ 2π ]+1
z s−1 ηs iη s−1 dz + + ln(1 − eiη ) z s(2π)s (2π)s Ctη e − 1 1 its−1 e−it t it − ln(1 − e ) + O + , (2π)s t2−σ η 2−σ (1 + )t < η < ∞, 0 ≤ σ ≤ 1, t → ∞,
where the error term is uniform for all η, σ in the above ranges. Proof. Letting σ → 1 − σ in (1.4) and taking the complex conjugate of the resulting equation, we ﬁnd s 1 t its−1 e−it n − + (−eit + 2i Im Li1 (eit )) ζ(1 − s) = s s 2π (2π) n=1 √ 1 + i π 6σ 2 − 6σ + 1 i ts−1 e−it 1 − i √ √ πt + − iσ − + +O(tσ−2 ). (2π)s 2 3 24 t t [ 2π ]
s−1
97
98
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
The lemma follows by subtracting this equation from (10.1), employing the following equations which follow from (3.13), (3.33), and (3.26) respectively: t iη s−1 iη ln(1 − e ) + O 2−σ , (1 + )t < η, (not uniformly in ), GL (t, σ; η) = − (2π)s η √ s−1 −it √ 1−i 1 + i π 6σ 2 − 6σ + 1 t e i (1) √ πt + − iσ − +O(tσ−2 ), GU (t, σ; t) = (2π)s 2 3 24 t its−1 e−it (2) ln(1 − e−it ) + O(tσ−2 ), GU (t, σ; t) = (2π)s and using the identity ts−1 e−it its−1 e−it its−1 e−it ln(1 − e−it ) + 2 Im Li1 (eit )) = ln(1 − eit ). s s (2π) (2π) (2π)s
It is possible to derive an alternative integral representation for the related sum [t/2π] s−1 . [η/2π]+1 n Lemma 10.2 (An alternative integral representation for the basic sum). For every δ > 0, (10.3) t [ 2π ]
ns−1 =
η n=[ 2π ]+1
0 < η < t,
t ρs−1 2 dρ + O(tσ−1 ), s iρ (2π) η e − 1
dist(η, 2πZ) > δ,
dist(t, 2πZ) > δ,
0 ≤ σ ≤ 1,
t → ∞,
where the error term is uniform for all η, σ in the above ranges and the contour in the integral denotes the principal value integral with respect to the points
η t 2πn n ∈ Z, +1≤n≤ . 2π 2π Proof. Let Cˆtη denote the semicircle from it to iη with Re z ≤ 0, deﬁned in equation (2.10c) with η and α replaced by t and η respectively, see ﬁgure 10.1. Let Rηt (s) and Lηt (s) denote the following: iπs z s−1 e− 2 t (10.4) dz, s∈C Rη (s) = (2π)s Cηt ez − 1 and e− 2 (2π)s iπs
(10.5)
Lηt (s) =
ˆη C t
z s−1 dz, ez − 1
s ∈ C.
Cauchy’s theorem applied in the interior of the disk whose boundary is Cηt ∪ Cˆtη , yields t [ 2π ]
(10.6)
Lηt (s)
+
Rηt (s)
=
ns−1 ,
s ∈ C.
η n=[ 2π ]+1
On the other hand, the Plemelj formulas imply (10.7)
Rηt (s) − Lηt (s) =
t ρs−1 2 dρ. (2π)s η eiρ − 1
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
99
it Cˆtη
Cηt iη Re z
Figure 10.1. The contours Cηt and Cˆtη . Using (10.7) to replace Rηt in (10.6), we ﬁnd t [ 2π ]
(10.8)
ns−1 =
η n=[ 2π ]+1
t ρs−1 2 dρ + 2Lηt . s iρ (2π) η e − 1
Replacing the contour Cˆtη in (10.5) by the union of the following three segments: {iteiθ  0 ≤ θ ≤ π/4}, we obtain Lηt
πi
{iρe 4  η ≤ ρ ≤ t},
{iηeiθ  0 ≤ θ ≤ π/4},
π iθ(s−1) s−1 t πi (s−1) s−1 4 e πi t e4 ρ 1 iθ e 4 dρ = dθ − ite πi iθ s ite (2π) e −1 0 η eiρe 4 − 1 π4 iθ(s−1) s−1 e η − iηeiθ dθ iθ iηe e −1 0 1 [J1 + J2 + J3 ]. =: (2π)s
The assumption dist(t, 2πZ) > δ implies that π4 −θt σ−1 πt e t tσ−1 J1  ≤ tdθ = (1 − e− 4 ) = O(tσ−1 ). A A 0 Similar computations show that J2 is exponentially small and, in view of the assumption dist(η, 2πZ) > δ, that J3 is O(η σ t−1 ). Thus, (10.9)
Lηt = O(tσ−1 ),
and the lemma follows from (10.8).
Remark 10.3. Lemma 10.2 implies that the leading behavior of ζ(s) is characterized by the following integral: t
I(η, t, σ) =
η
ρs−1 dρ. eiρ − 1
100
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
Remark 10.4. The estimate (10.9) is a consequence of the fact that the integral appearing in the deﬁnition of Lηt (s) does not possess any stationary points. Indeed, using ∞ 1 emz , Re z < 0, = − ez − 1 m=0 it follows that − iπs 2
e
ˆη C t
∞ t z s−1 dz = eimρ+it ln ρ ρσ−1 dρ. ez − 1 η m=0
Candidates for stationary points occur at ρ∗ = −t/m and the inequality η ≤ ρ∗ ≤ t implies the nonexistence of any stationary points. The next lemma gives an alternative representation for the fundamental integral Rηt (s) deﬁned in (10.4). Lemma 10.5. Let Cηt denote the semicircle from iη to it with Re z ≥ 0 deﬁned by equation ( 1.23). Then, for every δ > 0, (10.10) Rηt (s)
t e −1 us−1 du +O , iue−i − 1 t1−σ η e dist(η, 2πZ) > δ, dist(t, 2πZ) > δ,
e−is = (2π)s
0 < η < t,
t
0 ≤ σ ≤ 1,
0 < < 1,
t → ∞,
where the error term is uniform for all η, σ, in the above ranges. Proof. Let ∈ (0, 1). We deform the contour Cηt on the lhs of (10.10) so that it consists of the following three pieces: {iηe−iθ  0 ≤ θ ≤ },
{iue−i  η ≤ u ≤ t},
{ite−iθ  0 ≤ θ ≤ }.
This yields t −iθs −iθs iπs z s−1 dz us−1 du e dθ e dθ −is s s − iη = e + it . e− 2 −i −iθ −iθ z iue iηe ite −1 −1 −1 Cηt e − 1 η e 0 e 0 e The assumption that dist(η, 2πZ) > δ implies that there exists an A such that −iθ eiηe − 1 ≥ A for all θ ∈ [0, ], ∈ (0, 1), and η > 0. Thus, e−iθs eθt θ ∈ [0, ], eiηe−iθ − 1 ≤ A , and so
0
et − 1 e−iθs ≤ . dθ −iθ iηe At e −1
Similarly, because of the assumption dist(t, 2πZ) > δ, et − 1 e−iθs ≤ . dθ ite−iθ − 1 At 0 e This proves the lemma.
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
it
101
z∗
t−η
iη
Figure 10.2. The contour of integration of the integral in the rhs of equation (10.14). Lemma 10.5 implies that (10.11)
Rηt (s) =
1 lim (2π)s →0
t→0
η
t
us−1 du , eiue−i − 1
which in view of (10.6) and (10.9) implies the following alternative representation for the basic sum: t [ 2π t ] us−1 du 1 = lim ns−1 + O(tσ−1 ). −i →0 η eiue (2π)s t→0 −1 η n=[ 2π ]+1
In fact, in the remaining part of this chapter we will present arguments which suggest1 that on the critical line the assumption t → 0 in (10.11) can be relaxed to 2 t → 0, so that the following equation is valid: t us−1 du 1 (10.12) , lim Rηt (s) = s iue (2π) 2→0 η e −i − 1 t→0
where it is assumed that dist(η, 2πZ) > δ, η = O(1), dist(t, 2πZ) > δ, and σ = 1/2. Indeed, consider the integral e−z z s−1 dz s t − iπs 2 (2π) Rη (s) = e . 1 − e−z Cηt Instead of the contour Cηt , we consider the ﬁnite ray from the point iη to the point z ∗ , see ﬁgure 10.2, where π iπ π z ∗ = iη + (t − η)ei( 2 −) = [η + (t − η)e−i ]e 2 , 0≤ < . 2 We claim that it −z s−1 3 iπs e z dz e− 2 (10.13) = O(tσ et ). −z 1 − e ∗ z 1 The
rigorous justiﬁcation of some of these arguments remains open.
102
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
Indeed, the assumption dist(t, 2πZ) > δ implies that there exists an A > 0 independent of η, such that 1 − e−z  > A on the contour from z ∗ to it. Thus, using the parametrization z = ite−iθ , 0 ≤ θ ≤ , in the lhs of (10.13), we can prove (10.13) as follows: −t sin θ θt − iπs it e−z z s−1 dz s e−ite−iθ e−iθs dθ e e dθ σ e 2 = it ≤ t −iθ −z −ite 1−e A 1−e z∗ 0 0 3 3 = O tσ etθ dθ = O(tσ et ). 0
where we have used that 0 ≤ θ ≤ 1.
θ − sin θ ≤ θ 3 ,
It follows from (10.13) that z∗ −z s−1 iπs e−z z s−1 dz e z − iπs 2 e− 2 (10.14) = e lim dz. iπ →0 1 − e−z 1 − e−z 2 Cηt ηe 2 t→0
We next employ the following parametrization which maps z ∗ to ρ = 0: π iπ η iπ η z = ηe 2 + t 1 − − ρ ei( 2 −) = η + t 1 − − ρ e−i e 2 , t t η 0 0. η η Then, the rhs of equation (10.14) is given by 1− ηt −i eit(ρ−1)e ρ −iψ s−1 s −i s−1 −iμ(s−1) e (10.17a) t lim e M e dρ, 1 − −i →0 M 1 − eit(ρ−1)e 0 2 t→0
where (10.17b)
ψ(t, ) = − μ.
The functions M and μ are given by 2 η η2
2 η η 2 4 M (t, ) = 1 − − 2 +O
− 2 (10.18a) , 2 t t t t
→0
and
η + O( 2 ),
→ 0. t Indeed, consider the triangle D1 formed by the intersection of the following ﬁnite
(10.18b)
μ(t, ) = −
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
0
103
1 μ
t η
t ηM
−1
Figure 10.3. The triangle D1 . rays, see ﬁgure 10.3: (0, 1),
t 1, 1 + − 1 e−i , η
t 0, M e−iμ . η
The cosine rule for the triangle D1 implies the following identities: 2 2 t t t =1+ M −1 −2 − 1 cos(π − ) η η η 2 t t −1 +2 − 1 cos
=1+ η η 2 t t = 1+ −1 − 1 (1 − cos ). −2 η η Thus,
* + + t 2 − 1 (1 − cos ) η t t+ M = ,1 − , 2 t η η η2
or
M=
1−2
η η2 − 2 t t
(1 − cos ),
which yields (10.18a). The sine rule for the triangle D1 implies the identity sin μ sin(π − ) = . t −1 ηM
t η
Hence, 1 − ηt sin , M which, using the expression (10.18a) for M , yields (10.18b). Equation (10.18b) implies that ψ > 0, thus, we can deﬁne (W, w) via the equation ρ i(π−ψ) ρ −iψ (10.19) 1 − = 1+ = W eiw , W (ρ, t, ) > 0, w(ρ, t, ) > 0. e e M M The functions W and w are given by 2 2 2
η
η W (ρ, t, ) = 1 − ρ + O (10.20a) ,
→0 +O t t2 sin μ =
104
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
ρ M
W
ψ
w 0
1 Figure 10.4. The triangle D2 .
and (10.20b)
w(ρ, t, ) =
2 2 2
η
η
η ρ , 1+O +O t 1−ρ t t2
→ 0.
Indeed, consider the triangle D2 formed by the intersection of the following ﬁnite rays, see ﬁgure 10.4: ρ i(π−ψ) (0, 1), 1, 1 + , 0, W eiw . e M The cosine rule for the triangle D2 implies the following identities: ρ2 ρ W 2 = 1 + 2 − 2 cos ψ M M ρ 2 2ρ = 1− (1 − cos ψ). + M M Hence, ρ W = 1− M

2ρ 1 + Mρ 2 (1 − cos ψ). 1− M
Thus, using the deﬁnition (10.17b) of ψ, together with equation (10.18b), we ﬁnd ρ
ρ 1 M W =1− + ρ M 21− M
2 η 2 4 + O( ) , t2
→ 0.
Substituting in this equation the expression (10.18a) for M , we ﬁnd equation (10.20a). The sine rule for the triangle D2 implies the identity sin w sin ψ = , ρ/M W which, using the expressions for ψ (equations (10.17b) and (10.18b)), for M (equation (10.18a)) and for W (equation (10.20a)), yields equation (10.20b). Using the identity ρ −iψ s−1 e 1− = W s−1 ei(s−1)w , M
10. SEVERAL REPRESENTATIONS FOR THE BASIC SUM
105
as well as the expressions for M , μ, W and w obtained earlier, it is possible to compute all the expressions appearing in the rhs of (10.17a): 2 2 i2 η M s−1 = e(it+σ−1) ln M = e− 2 eO t eO t2 , η 2 2 e−i(s−1)μ = eμt e−i(σ−1)μ = et e−η eO( t) ei(σ−1)( t −) eO( ) , 2 η 2 2 2 η 2 η 2 s−1 s−1 (it+σ−1) O t +O t2 W = (1 − ρ) e = (1 − ρ)s−1 eO( η) eO t , ρ
ρ
eiw(s−1) = e−tw ei(σ−1)w = e−η 1−ρ e 1−ρ [O()+O( t )] .
Substituting the above expressions in (10.17a) and using the identities −i
e−ite
= e−it(1−i+O(
2
))
= e−it−t+O(
2
t)
,
in order to simplify the integrand of the integral in equation (10.17a), we ﬁnd the equation (10.21) 1− ηt it(ρ−1)e−i iπs e−z z s−1 dz e (1 − ρ)s−1 dρ s = t lim , 0 < η < t. e− 2 −i −z it(ρ−1)e →0 1−e 1−e Cηt 0 2 t→0
The transformation ρ = 1 − ut maps this equation to equation (10.12). We note that a detailed analysis in the neighborhood of ρ = 1 shows that similar estimates are also valid near ρ = 1.
APPENDIX A
The Asymptotics of Γ(1 − s) and χ(s) The asymptotic formulae given in theorems 4.1 and 4.4 involve the functions χ(s) and Γ(1 − s). In this appendix, we derive the asymptotics of these functions as t → ∞. We use the following wellknown formula for the asymptotics of the Gamma function (see for example Olver [20], page 294): −s s ln s
Γ(s) = e
e
2π s
12 1+
1 +O 12s
1 s2
where δ > 0. Hence, as t → ∞, Γ(s) =
√
2πe−σ−it es ln[(it)(1+ it )] σ
1+
1 σ 12it(1+ it )
1 (it) 2 1 +
s → ∞,
,
+O 1 σ 2
1 s2
 arg s ≤ π − δ,
it √ s σ σ2 1 2π iπ (σ+it) it + 2t2 +O ( t3 ) − iπ = √ e 4 e 2 t e−σ−it e t 1 1 1 σ +O 2 +O 2 × 1+ (A.1) 1− 12it t 2it t √ s− 1 iπs − iπ −it − iσ2 +O( 1 ) 1 1 − 6σ 2 2t 2 2 4 t +O 2 e e e e = 2πt 1+ 12it t 2 √ s− 1 iπs − iπ −it 1 1 iσ 1 − 6σ 2 2 4 +O 2 +O 2 = 2πt e e e 1− 1+ 2t t 12it t √ s− 1 iπs − iπ −it 1 ic(σ) = 2πt 2 e 2 e 4 e +O 2 1+ , 0 ≤ σ ≤ 1, t → ∞, t t
where c(σ) is deﬁned by 1 σ (1 − σ) − . 2 12 Replacing σ by 1 − σ in (A.1) and taking the complex conjugate of the resulting equation, we ﬁnd √ −iπ(1−s) 1 iπ 1 ic(σ) +O 2 e 4 eit 1 − , Γ(1 − s) = 2πt 2 −s e 2 t t (A.2) 0 ≤ σ ≤ 1, t → ∞. c(σ) =
On the other hand, the deﬁnition (2.2) of χ(s) implies (2π)s χ(1 − s) =
iπ iπs iπ iπs (2π)s (2π)1−s Γ(s) e 2 (1−s) − e− 2 (1−s) = e− 2 + e 2 Γ(s). 2i π 107
A. THE ASYMPTOTICS OF Γ(1 − s) AND χ(s)
108
Replacing in this equation Γ(s) by the rhs of (A.1) we ﬁnd (A.3) χ(1 − s) = (2π)
1 2 −s
t
s− 12 − iπ −it 4
e
e
1 ic(σ) +O 2 1+ , t t
0 ≤ σ ≤ 1,
t → ∞.
Equation (A.3) together with the identity (2.13) imply the following asymptotic expression for χ(s): 1 1 iπ 1 ic(σ) , 0 ≤ σ ≤ 1, t → ∞. χ(s) = (2π)s− 2 t 2 −s e 4 eit 1 − +O 2 t t It is of course straightforward to extend the above asymptotic formulae to higher order.
APPENDIX B
Numerical Veriﬁcations B.1. Veriﬁcation of Theorem 3.1 Letting σ = 1/2, the error term in equation (3.2) is given by t = 10 −(10.4 + 5.22i) × 10−5 −(4.00 + 4.19i) × 10−3
2
η=t 3 η = t2
t = 102 (−10.2 + 2.97i) × 10−9 −(1.40 + 1.11i) × 10−5
t = 103 (15.1 − 4.46i) × 10−13 −(7.80 + 9.81i) × 10−8
Note that the error is proportional to t3 /η 3+σ as expected. In order to demonstrate the eﬀect of the higher order terms in (3.2), we also consider the diﬀerence between ζ(s) and the ﬁrst term on the rhs of (3.2), i.e., the diﬀerence η [ 2π ] n−s . ζ(s) − n=1
For σ = 1/2, this diﬀerence is given by 2
η=t 3 η = t2
t = 10 t = 102 t = 103 −0.291 + 0.274i −0.341 + 0.207i −0.380 + 0.121i 0.266 + 0.0471i 0.127 + 0.020i 0.0360 + 0.0612i
Similarly, the diﬀerence between ζ(s) and the ﬁrst two terms on the rhs of (3.2), i.e., the diﬀerence [ η ] 2π 1 η 1−s −s ζ(s) − n − , 1 − s 2π n=1 is given by 2
η=t 3 η = t2
t = 10 −(8.27 + 6.52i) × 10−2 (1.58 − 1.50i) × 10−1
t = 102 −(6.95 + 10.3i) × 10−4 (9.37 − 26.0i) × 10−3
t = 103 −(3.40 + 10.6i) × 10−4 (−4.93 + 3.32i) × 10−3
The above example illustrates that the inclusion of the higher order terms in (3.2) improves the convergence of the asymptotic series considerably. B.2. Veriﬁcation of Theorem 3.2 The error term in equation (3.22) is given by σ=0 σ = 1/2 σ=1
t = 10 (1.97 − 3.81i) × 10−3 (2.23 − 4.34i) × 10−3 (2.42 − 4.78i) × 10−3
t = 102 (−7.76 + 65.1i) × 10−7 (−2.74 + 23.5i) × 10−7 (−9.41 + 82.5i) × 10−8
t = 103 (5.62 − 3.40i) × 10−9 (6.46 − 3.82i) × 10−10 (7.13 − 4.22i) × 10−11
Note that the error is proportional to t−σ−3 as expected. 109
110
B. NUMERICAL VERIFICATIONS
B.3. Veriﬁcation of Theorem 4.1 Letting σ = 1/2, the error term in equation (4.2), i.e., the term 1 1 < η < t 3 < ∞, O ηt , −iπs − πt σ−1 2 e Γ(1 − s)e η × √ − At 4 1 O e η2 + ηt2 , t 3 < η < t < ∞, is given by t = 102 (3.04 + 7.27i) × 10−3 (45.4 − 6.75i) × 10−5 7.27 − 1.45i) × 10−1
η = 10 η = t1/4 η = t5/12
t = 104 (8.05 − 2.53i) × 10−6 (8.05 − 2.53i) × 10−6 (−8.69 + 9.53i) × 10−5
t = 106 (84.1 − 5.12i) × 10−10 (−443.6 + 6.91i) × 10−7 (4.38 − 13.7i) × 10−6
In order to illustrate the increased accuracy achieved by including the higher order terms in (4.2), we also display the corresponding table when only the ﬁrst two sums on the rhs of (4.2) are included: For σ = 1/2, the diﬀerence1 [t]
η
[ 2π ] η 1 1 ζ(s) − − χ(s) s n n1−s n=1 n=1
is given by t = 102 (5.55 − 14.1i) × 10−2 (4.76 − 7.53i) × 10−2 −(2.14 + 2.15i) × 10−1
η = 10 η = t1/4 η = t5/12
t = 104 (−14.4 + 7.92i) × 10−3 (−14.4 + 7.92i) × 10−3 (3.09 − 1.91i) × 10−2
t = 106 (−15.9 + 4.40i) × 10−4 −(26.3 + 6.92i) × 10−3 (−5.70 + 10.4i) × 10−3
B.4. Veriﬁcation of Corollary 4.3 Letting σ = 1/2 and N = 2, we ﬁnd that the error term in equation (4.24) is given by 7 12
η=t 3 η = t4 η = 10t
t = 10 −(4.08 + 2.47i) × 10−1 (6.92 + 55.7i) × 10−2 (1.40 − 1.12i) × 10−2
t = 102 (−2.39 + 3.50i) × 10−1 (9.93 + 25.7i) × 10−2 (2.24 + 5.28i) × 10−4
t = 103 (−6.42 + 15.9i) × 10−2 −(8.75 + 12.2i) × 10−3 (2.45 + 186.1i) × 10−7
B.5. Veriﬁcation of Theorem 4.4 Letting σ = 1/2, the error term in equation (4.31), i.e., the term 1 1 < η < t 3 < ∞, O ηt , −iπs − πt σ−1 Γ(1 − s)e 2 η × e √ − At 4 1 O e η2 + ηt2 , t 3 < η < t < ∞, is given by η=
2πt √ 100
η = 2πt √ η = 200πt
t = 102
t = 104
(−5.96 + 8.83i) ×
10−4
10−3
(3.59 − 10.8i) × (13.7 − 5.21i) × 10−6
(6.44 + 1.30i) ×
t = 106 10−4
(4.64 + 69.7i) × 10−7
10−5
(104.6 + 8.97i) × 10−7 −(14.2 + 6.74i) × 10−7
(2.72 − 28.1i) × (−9.02 + 2.27i) × 10−4
1 This diﬀerence is exactly the error in the “approximate functional equation” of Hardy and Littlewood cf. [12].
B.5. VERIFICATION OF THEOREM 4.4
111
√ Remark B.1. In the particular case of η = 2πt, the formula of Siegel (Eq. (32) of [24]) and the corresponding formula of Titchmarsh (Theorem 4.16 of [26]) are equivalent to the formula of our theorem 4.4. This equivalence can easily be checked numerically—for example, for N = 3 all three formulas yield the same numerical values for the error terms. In this regard, we note that Titchmarsh states Siegel’s formula in terms of the function Ψ(a) deﬁned by 2 iw2 a2 5 cos π( a2 − a − 18 ) e 4π + aw e−iπ( 2 − 8 ) Ψ(a) = dw = , 2π ew − 1 cos πa L which is related to our function Φ(τ, u) deﬁned in (4.29) by 1 −iπ( a22 − 58 ) e Ψ(a) = −iΦ −1, a − , a ∈ C. 2
Bibliography [1] J. Andersson, Mean value properties of the Hurwitz zetafunction, Math. Scand. 71 (1992), no. 2, 295–300, DOI 10.7146/math.scand.a12430. MR1212712 [2] A. C. L. Ashton and A. S. Fokas, Relations among the Riemann zeta and Hurwitz zeta functions as well as their products, preprint. [3] E. P. Balanzario and J. S´ anchezOrtiz, RiemannSiegel integral formula for the Lerch zeta function, Math. Comp. 81 (2012), no. 280, 2319–2333, DOI 10.1090/S002557182011025664. MR2945158 [4] R. Balasubramanian, A note on Hurwitz’s zetafunction, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 41–44, DOI 10.5186/aasfm.197879.0401. MR538086 [5] M. V. Berry, The RiemannSiegel expansion for the zeta function: high orders and remainders, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1939, 439–462, DOI 10.1098/rspa.1995.0093. MR1349513 [6] M. V. Berry and J. P. Keating, A new asymptotic representation for ζ( 12 + it) and quantum spectral determinants, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 151–173, DOI 10.1098/rspa.1992.0053. MR1177749 [7] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of Lfunctions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104, DOI 10.1112/S0024611504015175. MR2149530 [8] H. M. Edwards, Riemann’s zeta function, Dover Publications, Inc., Mineola, NY, 2001. Reprint of the 1974 original [Academic Press, New York; MR0466039 (57 #5922)]. MR1854455 [9] A. Fernandez and A. S. Fokas, Asymptotics to all orders of the Hurwitz zeta function, J. Math. Anal. Appl. 465 (2018), no. 1, 423–458, DOI 10.1016/j.jmaa.2018.05.012. MR3806712 [10] R. Garunkshtis, A. Laurinchikas, and I. Steuding, An approximate functional equation for the Lerch zeta function (Russian, with Russian summary), Mat. Zametki 74 (2003), no. 4, 494–501, DOI 10.1023/A:1026183524830; English transl., Math. Notes 74 (2003), no. 34, 469–476. MR2042962 [11] R. Garunkˇstis, A. Laurinˇ cikas, and J. Steuding, On the mean square of Lerch zetafunctions, Arch. Math. (Basel) 80 (2003), no. 1, 47–60, DOI 10.1007/s000130300005. MR1968287 [12] G. H. Hardy and J. E. Littlewood, The Approximate Functional Equations for zeta(s) and zeta2(s), Proc. London Math. Soc. (2) 29 (1929), no. 2, 81–97, DOI 10.1112/plms/s229.1.81. MR1575331 [13] K. Kalimeris and A. S. Fokas, Explicit asymptotics for certain single and double exponential sums, preprint, arXiv:1708.02868. [14] M. Katsurada and K. Matsumoto, Discrete mean values of Hurwitz zetafunctions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 6, 164–169. MR1232817 [15] M. Katsurada and K. Matsumoto, Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zetafunctions. I, Math. Scand. 78 (1996), no. 2, 161–177, DOI 10.7146/math.scand.a12580. MR1414645 [16] J. P. Keating, In Quantum chaos (eds. G. Casati, I. Guarneri, & U. Smilansky), Amsterdam: NorthHolland, 1993, pp. 145–185. [17] J. F. Koksma and C. G. Lekkerkerker, A meanvalue theorem for ζ(s, w), Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14 (1952), 446–452. MR0058636 [18] T. Miyagawa, Approximate functional equations for the Hurwitz and Lerch zetafunctions, Comment. Math. Univ. St. Pauli 66 (2017), no. 12, 15–27. MR3752435
113
114
BIBLIOGRAPHY
[19] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.20 of 20180915. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. [20] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1974. Computer Science and Applied Mathematics. MR0435697 [21] V. V. Rane, On Hurwitz zeta function, Math. Ann. 264 (1983), no. 2, 147–151, DOI 10.1007/BF01457521. MR711874 [22] V. V. Rane, A new approximate functional equation for Hurwitz zeta function for rational parameter, Proc. Indian Acad. Sci. Math. Sci. 107 (1997), no. 4, 377–385, DOI 10.1007/BF02837221. MR1484372 ¨ [23] B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebenen Gr¨ oße, Monatsberichte der K¨ oniglichen Preußischen Akademie der Wissenschaften zu Berlin (1859), 671–680. ¨ [24] C. L. Siegel, Uber Riemanns Nachlaß zur analytischen Zahlentheorie, Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2: 4580 (1932), reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: SpringerVerlag, 1966. [25] K. Soundararajan, Moments of the Riemann zeta function, Ann. of Math. (2) 170 (2009), no. 2, 981–993, DOI 10.4007/annals.2009.170.981. MR2552116 [26] E. C. Titchmarsh, The theory of the Riemann zetafunction, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. HeathBrown. MR882550 [27] W. P. Zhang, On the Hurwitz zetafunction, Northeast. Math. J. 6 (1990), no. 3, 261–267. MR1078941 [28] W. P. Zhang, On the Hurwitz zetafunction, Illinois J. Math. 35 (1991), no. 4, 569–576. MR1115987 [29] W. P. Zhang, On the mean square value of the Hurwitz zetafunction, Illinois J. Math. 38 (1994), no. 1, 71–78. MR1245834
Editorial Information To be published in the Memoirs, a paper must be correct, new, nontrivial, and signiﬁcant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers appearing in Memoirs are generally at least 80 and not more than 200 published pages in length. Papers less than 80 or more than 200 published pages require the approval of the Managing Editor of the Transactions/Memoirs Editorial Board. Published pages are the same size as those generated in the style ﬁles provided for AMSLATEX. Information on the backlog for this journal can be found on the AMS website starting from http://www.ams.org/memo. A Consent to Publish is required before we can begin processing your paper. After a paper is accepted for publication, the Providence oﬃce will send a Consent to Publish and Copyright Agreement to all authors of the paper. By submitting a paper to the Memoirs, authors certify that the results have not been submitted to nor are they under consideration for publication by another journal, conference proceedings, or similar publication. Information for Authors Memoirs is an authorprepared publication. Once formatted for print and online publication, articles will be published as is with the addition of AMSprepared frontmatter and backmatter. Articles are not copyedited; however, conﬁrmation copy will be sent to the authors. Initial submission. The AMS uses Centralized Manuscript Processing for initial submissions. Authors should submit a PDF ﬁle using the Initial Manuscript Submission form found at www.ams.org/submission/memo, or send one copy of the manuscript to the following address: Centralized Manuscript Processing, MEMOIRS OF THE AMS, 201 Charles Street, Providence, RI 029042294 USA. If a paper copy is being forwarded to the AMS, indicate that it is for Memoirs and include the name of the corresponding author, contact information such as email address or mailing address, and the name of an appropriate Editor to review the paper (see the list of Editors below). The paper must contain a descriptive title and an abstract that summarizes the article in language suitable for workers in the general ﬁeld (algebra, analysis, etc.). The descriptive title should be short, but informative; useless or vague phrases such as “some remarks about” or “concerning” should be avoided. The abstract should be at least one complete sentence, and at most 300 words. Included with the footnotes to the paper should be the 2020 Mathematics Subject Classiﬁcation representing the primary and secondary subjects of the article. The classiﬁcations are accessible from www.ams.org/msc/. The Mathematics Subject Classiﬁcation footnote may be followed by a list of key words and phrases describing the subject matter of the article and taken from it. Journal abbreviations used in bibliographies are listed in the latest Mathematical Reviews annual index. The series abbreviations are also accessible from www.ams.org/msnhtml/serials.pdf. To help in preparing and verifying references, the AMS oﬀers MR Lookup, a Reference Tool for Linking, at www.ams.org/mrlookup/. Electronically prepared manuscripts. The AMS encourages electronically prepared manuscripts, with a strong preference for AMSLATEX. To this end, the Society has prepared AMSLATEX author packages for each AMS publication. Author packages include instructions for preparing electronic manuscripts, samples, and a style ﬁle that generates the particular design speciﬁcations of that publication series. Authors may retrieve an author package for Memoirs of the AMS from www.ams.org/ journals/memo/memoauthorpac.html. The AMS Author Handbook is available in PDF format from the author package link. The author package can also be obtained free of charge by sending email to [email protected] or from the Publication Division,
American Mathematical Society, 201 Charles St., Providence, RI 029042294, USA. When requesting an author package, please specify the publication in which your paper will appear. Please be sure to include your complete mailing address. After acceptance. The source ﬁles for the ﬁnal version of the electronic manuscript should be sent to the Providence oﬃce immediately after the paper has been accepted for publication. The author should also submit a PDF of the ﬁnal version of the paper to the editor, who will forward a copy to the Providence oﬃce. Accepted electronically prepared ﬁles can be submitted via the web at www.ams.org/ submitbookjournal/, sent via FTP, or sent on CD to the Electronic Prepress Department, American Mathematical Society, 201 Charles Street, Providence, RI 029042294 USA. TEX source ﬁles and graphic ﬁles can be transferred over the Internet by FTP to the Internet node ftp.ams.org (130.44.1.100). When sending a manuscript electronically via CD, please be sure to include a message indicating that the paper is for the Memoirs. Electronic graphics. Comprehensive instructions on preparing graphics are available at www.ams.org/authors/journals.html. A few of the major requirements are given here. Submit ﬁles for graphics as EPS (Encapsulated PostScript) ﬁles. This includes graphics originated via a graphics application as well as scanned photographs or other computergenerated images. If this is not possible, TIFF ﬁles are acceptable as long as they can be opened in Adobe Photoshop or Illustrator. Authors using graphics packages for the creation of electronic art should also avoid the use of any lines thinner than 0.5 points in width. Many graphics packages allow the user to specify a “hairline” for a very thin line. Hairlines often look acceptable when proofed on a typical laser printer. However, when produced on a highresolution laser imagesetter, hairlines become nearly invisible and will be lost entirely in the ﬁnal printing process. Screens should be set to values between 15% and 85%. Screens which fall outside of this range are too light or too dark to print correctly. Variations of screens within a graphic should be no less than 10%. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Managing Editor and the Publisher. In general, color graphics will appear in color in the online version. Inquiries. Any inquiries concerning a paper that has been accepted for publication should be sent to [email protected] or directly to the Electronic Prepress Department, American Mathematical Society, 201 Charles St., Providence, RI 029042294 USA.
Editors This journal is designed particularly for long research papers, normally at least 80 pages in length, and groups of cognate papers in pure and applied mathematics. Papers intended for publication in the Memoirs should be addressed to one of the following editors. The AMS uses Centralized Manuscript Processing for initial submissions to AMS journals. Authors should follow instructions listed on the Initial Submission page found at www.ams.org/memo/memosubmit.html. Managing Editor: Henri Darmon, Department of Mathematics, McGill University, Montreal, Quebec H3A 0G4, Canada; email: [email protected] 1. GEOMETRY, TOPOLOGY & LOGIC Coordinating Editor: Richard Canary, Department of Mathematics, University of Michigan, Ann Arbor, MI 481091043 USA; email: [email protected] Algebraic topology, Michael Hill, Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095 USA; email: [email protected] Logic, Mariya Ivanova Soskova, Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706 USA; email: [email protected] Lowdimensional topology and geometric structures, Richard Canary Symplectic geometry, Yael Karshon, School of Mathematical Sciences, TelAviv University, Tel Aviv, Israel; and Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada; email: [email protected] 2. ALGEBRA AND NUMBER THEORY Coordinating Editor: Henri Darmon, Department of Mathematics, McGill University, Montreal, Quebec H3A 0G4, Canada; email: [email protected] Algebra, Radha Kessar, Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom; email: [email protected] Algebraic geometry, Lucia Caporaso, Department of Mathematics and Physics, Roma Tre University, Largo San Leonardo Murialdo, I00146 Rome, Italy; email: [email protected] Analytic number theory, Lillian B. Pierce, Department of Mathematics, Duke University, 120 Science Drive Box 90320, Durham, NC 27708 USA; email: [email protected] Arithmetic geometry, Ted C. Chinburg, Department of Mathematics, University of Pennsylvania, Philadelphia, PA 191046395 USA; email: [email protected] Commutative algebra, Irena Peeva, Department of Mathematics, Cornell University, Ithaca, NY 14853 USA; email: [email protected] Number theory, Henri Darmon 3. GEOMETRIC ANALYSIS & PDE Coordinating Editor: Alexander A. Kiselev, Department of Mathematics, Duke University, 120 Science Drive, Rm 117 Physics Bldg, Durham, NC 27708 USA; email: [email protected] Diﬀerential geometry and geometric analysis, Ailana M. Fraser, Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Room 121, Vancouver BC V6T 1Z2, Canada; email: [email protected] Harmonic analysis and partial diﬀerential equations, Monica Visan, Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095 USA; email: [email protected] Partial diﬀerential equations and functional analysis, Alexander A. Kiselev Real analysis and partial diﬀerential equations, Joachim Krieger, Bˆ atiment de Math´ ematiques, ´ Ecole Polytechnique F´ ed´ erale de Lausanne, Station 8, 1015 Lausanne Vaud, Switzerland; email: [email protected] 4. ERGODIC THEORY, DYNAMICAL SYSTEMS & COMBINATORICS Coordinating Editor: Vitaly Bergelson, Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210 USA; email: [email protected] Algebraic and enumerative combinatorics, Jim Haglund, Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 USA; email: [email protected] Probability theory, Robin Pemantle, Department of Mathematics, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104 USA; email: [email protected] Dynamical systems and ergodic theory, Ian Melbourne, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom; email: [email protected] Ergodic theory and combinatorics, Vitaly Bergelson 5. ANALYSIS, LIE THEORY & PROBABILITY Coordinating Editor: Stefaan Vaes, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B3001 Leuven, Belgium; email: [email protected] Functional analysis and operator algebras, Stefaan Vaes Harmonic analysis, PDEs, and geometric measure theory, Svitlana Mayboroda, School of Mathematics, University of Minnesota, 206 Church Street SE, 127 Vincent Hall, Minneapolis, MN 55455 USA; email: [email protected] Probability theory and stochastic analysis, Davar Khoshnevisan, Department of Mathematics, The University of Utah, Salt Lake City, UT 84112 USA; email: [email protected]
SELECTED PUBLISHED TITLES IN THIS SERIES
1342 Haosui Duanmu, Jeﬀrey S. Rosenthal, and William Weiss, Ergodicity of Markov Processes via Nonstandard Analysis, 2018 1341 Mima Stanojkovski, Intense Automorphisms of Finite Groups, 2021 1340 Le Chen, Yaozhong Hu, and David Nualart, Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations, 2021 1339 Gong Chen, Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians, 2021 1338 Andr´ as Juh´ asz, Dylan P. Thurston, and Ian Zemke, Naturality and Mapping Class Groups in Heegard Floer Homology, 2021 1323 Shu Kawaguchi and Kazuhiko Yamaki, Eﬀective Faithful Tropicalizations Associated to Linear Systems on Curves, 2021 1322 D. Bulacu and B. Torrecillas, Galois and Cleft Monoidal Cowreaths. Applications, 2021 1321 Christian Haase, Andreas Paﬀenholz, Lindsey C. Piechnik, and Francisco Santos, Existence of Unimodular Triangulations–Positive Results, 2021 1320 Th. Heidersdorf and R. Weissauer, Cohomological Tensor Functors on Representations of the General Linear Supergroup, 2021 1319 Abed Bounemoura and Jacques F´ ejoz, Hamiltonian Perturbation Theory for UltraDiﬀerentiable Functions, 2021 1318 Chao Wang, Zhifei Zhang, Weiren Zhao, and Yunrui Zheng, Local WellPosedness and BreakDown Criterion of the Incompressible Euler Equations with Free Boundary, 2021 1317 Eric M. Rains and S. Ole Warnaar, Bounded Littlewood Identities, 2021 1316 Ulrich Bunke and David Gepner, Diﬀerential Function Spectra, the Diﬀerential BeckerGottlieb Transfer, and Applications to Diﬀerential Algebraic KTheory, 2021 ´ Matheron, and Q. Menet, Linear Dynamical Systems on Hilbert 1315 S. Grivaux, E. Spaces: Typical Properties and Explicit Examples, 2021 1314 Pierre Albin, Fr´ ed´ eric Rochon, and David Sher, Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps, 2021 1313 Paul Godin, The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners, 2021 1312 Patrick Delorme, Pascale Harinck, and Yiannis Sakellaridis, PaleyWiener Theorems for a pAdic Spherical Variety, 2021 1311 Lyudmila Korobenko, Cristian Rios, Eric Sawyer, and Ruipeng Shen, Local Boundedness, Maximum Principles, and Continuity of Solutions to Inﬁnitely Degenerate Elliptic Equations with Rough Coeﬃcients, 2021 1310 Hiroshi Iritani, Todor Milanov, Yongbin Ruan, and Yefeng Shen, GromovWitten Theory of Quotients of Fermat CalabiYau Varieties, 2021 1309 J´ er´ emie Chalopin, Victor Chepoi, Hiroshi Hirai, and Damian Osajda, Weakly Modular Graphs and Nonpositive Curvature, 2020 1308 Christopher L. Douglas, Christopher SchommerPries, and Noah Snyder, Dualizable Tensor Categories, 2020 1307 Adam R. Thomas, The Irreducible Subgroups of Exceptional Algebraic Groups, 2020 1306 Kazuyuki Hatada, Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms, 2020 1305 Bogdan Ion and Siddhartha Sahi, Double Aﬃne Hecke Algebras and Congruence Groups, 2020
For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/memoseries/.
Memoirs of the American Mathematical Society
9 781470 450984
MEMO/275/1351
Number 1351 • January 2022
ISBN 9781470450984