Analytic Number Theory 2004045081

Table of contents :
Contents
Preface
Introduction
1 Arithmetic functions
2 Elementary theory of prime numbers
3 Characters
4 Summation formulas
5 Classical analytic theory of L-functions
6 Elementary sieve methods
7 Bilinear forms and the large sieve
8 Exponential sums
9 The Dirichlet polynomials
10 Zero-density estimates
11 Sums over finite fields
12 Character sums
13 Sums over primes
14 Holomorphic modular forms
15 Spectral theory of automorphic forms
16 Sums of Kloosterman sums
17 Primes in arithmetic progressions
18 The least prime in an arithmetic progression
19 The Goldbach problem
20 The circle method
21 Equidistribution
22 Imaginary quadratic fields
23 Effective bounds for the class number
24 The critical zeros of the Riemann zeta function
25 The spacing of zeros of the Riemann zeta-function
26 Central values of L-functions
Bibliography
Index

Citation preview

Analyti c Numbe r Theor y

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http://dx.doi.org/10.1090/coll/053

America n Mathematica l Societ y Colloquiu m Publication s Volum e 53

Analyti c Numbe r Theor y Henry k Iwanie c Emmanue l Kowalsk i

America n Mathematica l Societ y Providence , Rhod e Islan d

Editorial Boar d Susan J . Priedlander , Chai r Yuri Mani n Peter Sarna k 2000 Mathematics Subject

Classification.

P r i m a r y H F x x , H L x x , H M x x , H N x x , 1 1 T23 , 11T24, 1 1 R42 .

For a d d i t i o n a l informatio n a n d u p d a t e s o n t h i s book , visi t www.ams.org/bookpages/coll-53

Library o f Congres s Cataloging-in-Publicatio n D a t a Iwaniec, Henryk . Analytic numbe r theor y / Henry k Iwaniec , Emmanue l Kowalski . p. cm . — (Colloquiu m publications , ISS N 0065-925 8 ; v. 53 ) Includes bibliographica l reference s an d index . ISBN 0-821 8-3633- 1 (acid-fre e paper ) 1. Numbe r theory . I . Kowalski , Emmanuel , 1 969 - II . Title . III . Colloquiu m publication s (American Mathematica l Society ) ; v. 53 . QA241.I85 200 4 512.7'"3-dc22

2004045081

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o reprint-permissionQams.org . © 200 4 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t h t t p : //www. ams. o r g/ 10 9 8 7 6 5 4 1 3 2

51 41 31 21 1 1 0

Contents Preface x

i

Introduction Chapter 1 . Arithmeti c Function s 9 §1.1. Notatio n an d definition s 9 §1.2. Generatin g serie s §1.3. Dirichle t convolutio n §1.4. Example s §1.5. 1 Arithmeti c function s o n averag e §1.6. Sum s o f multiplicativ e function s 2 §1.7. Distributio n o f additiv e function s 2

0 2 3 9 3 8

Chapter 2 . Elementar y Theor y o f Prim e Number s 3 1 §2.1. Th e Prim e Numbe r Theore m 3 1 §2.2. Tchebyshe v metho d 3 2 §2.3. Prime s i n arithmeti c progression s 3 4 §2.4. Reflection s o n elementar y proof s o f th e Prim e Numbe r Theore m 3 8 Chapter 3 . Character s 4 §3.1. Introductio n 4 §3.2. Dirichle t character s 4 §3.3. Primitiv e character s 4 §3.4. Gaus s sum s 4 §3.5. Rea l character s 4 §3.6. Th e quarti c residu e symbo l 5 §3.7. Th e Jacobi-Dirichle t an d th e Jacobi-Kubot a symbol s 5 §3.8. Heck e character s 5

3 3 4 5 7 9 3 5 6

Chapter 4 . Summatio n Formula s 6 5 §4.1. Introductio n 6 5 §4.2. Th e Euler-Maclauri n formul a 6 6 §4.3. Th e Poisso n summatio n formul a 6 9 §4.4. Summatio n formula s fo r th e bal l 7 1 §4.5. Summatio n formula s fo r th e hyperbol a 7 4 §4.6. Functiona l equation s o f Dirichle t L-function s 8 4 §4.A. Appendix : Fourie r integral s an d serie s 86 Chapter 5 . Classica l Analyti c Theor y o f L- functions 9 §5.1. Definition s an d preliminarie s 9 v

3 3

vi CONTENT

S

§5.2. Approximation s t o L-function s 9 7 1 §5.3. Countin g zero s o f L-function s 0 1 §5.4. Th e zero-fre e regio n 0 5 §5.5. Explici t formul a 0 8 1 1 §5.6. Th e prim e numbe r theore m 0 §5.7.1 1 Th e Gran d Rieman n Hypothesi s 3 1 1 §5.8. Simpl e consequence s o f GR H 7 §5.9. Th e Rieman n zet a functio n an d Dirichle1 1 t L-function s 9 §5.10. L-function s o f numbe r fields 1 2 5 §5.11. Classica 1 l automorphi c L-function s 3 1 §5.12.1 Genera l automorphi c L-function s 3 6 §5.13. Arti n L-function s 4 1 1 §5.14. L-function s o f varietie s 4 5 1 §5.A. Appendix : comple x analysi s 4 9 Chapter 1 6 . Elementar y Siev e Method s 5 §6.1. Siev e problem s 5 1 §6.2. Exclusion-inclusio n schem e 5 1 §6.3. Estimation s o f V+(z), V~(z) 5 §6.4. Fundamenta 1 l Lemm a o f siev e theor y 5 §6.5. Th e A 2-Sieve 6 §6.6. Estimat e fo r th e mai 1 n ter m o f th e A 2-sieve 6 §6.7. Estimate s fo r th e remainde r ter1 m i n th e A 2-sieve 6 2 §6.8. 1 Selecte d application s o f A -sieve 6

3 3 4 7 8 0 4 5 6

Chapter 7 . Bilinea 1 r Form s an d th e Larg e Siev e 6 9 §7.1. Genera l principle s o f estimatin 1 g doubl e sum s 6 9 §7.2. 1 Bilinea r form s wit h exponential s 7 1 1 §7.3. Introductio n t o th e larg e siev e 7 4 §7.4. 1 Additiv e larg e siev e inequalitie s 7 5 §7.5. Multiplicativ 1 e larg e siev e inequalit y 7 9 §7.4. Application s o f th e larg e siev e t o 1 sievin g problem s 8 0 §7.6. Panoram a o f1 th e larg e siev e inequalitie s 8 3 §7.7. Larg e siev1 e inequalitie s fo r cus p form s 8 6 §7.8. 1 Orthogonalit y o f ellipti c curve s 9 2 1 §7.9. Powe r moment s o f L-function s 9 4 Chapter 8 . Exponentia l Sum s 9 §8.1. Introductio n 9 §8.2. Weyl' s metho d 9 §8.3. Va n de r Corpu t metho d 20 1 §8.4. Discussio n o f exponen t pair s 2 1 §8.5. Vinogradov' s metho d 2 Chapter 9 . Th e Dirichle t Polynomial s 22 §9.1. Introductio n 22 §9.2. Th e integra l mean-valu e estimate s 23 §9.3. Th e discret e mean-valu e estimate s 23 §9.4. Larg e value s o f Dirichle t polynomial s 23 §9.5. Dirichle t polynomial s wit h character s 23

7 7 8 4 3 6 9 9 0 2 5 8

TABLE O F CONTENT S vi

§9.6. Th e reflectio n metho d 24 §9.7. Larg e value s o f D(s, x) 24 Chapter 1 0 . Zer o Densit y Estimate s 24 §10.1. Introductio n 24 §10.2. Zero-detectin g polynomial s 25 §10.3. Breakin g th e zero-densit y conjectur e 25 §10.4. Gran d zero-densit y theore m 25 §10.5. Th e gap s betwee n prime s 26

i

3 6 9 9 0 4 6 4

Chapter 1 1 . Sum s ove r Finit e Field s 26 9 §11.1. Introductio n 26 9 §11.2. Finit e field s 26 9 §11.3. Exponentia l sum s 27 2 §11.4. Th e Hasse-Davenpor t relatio n 27 4 §11.5. Th e zet a functio n fo r Kloosterma n sum s 27 8 §11.6. Stepanov' s metho d fo r hyperellipti c curve s 28 1 §11.7. Proo f o f Weil' s boun d fo r Kloosterma n sum s 28 7 §11.8. Th e Rieman n Hypothesi s fo r ellipti c curve s ove r finit e field s 29 0 §11.9. Geometr y o f ellipti c curve s 29 1 §11.10. Th e loca l zet a functio n o f ellipti c curve s 29 7 §11.11. Surve y o f furthe r results : a cohomologica l prime r 30 0 §11.12. Comment s 3 3 Chapter 1 2 . Characte r Sum s 3 §12.1. Introductio n 3 §12.2. Completin g method s 3 1 §12.3. Complet e characte r sum s 3 §12.4. Shor t characte r sum s 32 §12.5. Ver y shor t characte r sum s t o highl y composit e modulu s 330 §12.6. Character s t o powerfu l modulu s 33 Chapter 1 3 . Sum s ove r Prime s 33 §13.1. Genera l principle s 33 §13.2. A varian t o f Vinogradov' s metho d 340 §13.3. Linnik' s identit y 34 §13.4. Vaughan' s identit y 34 §13.5. Exponentia l sum s ove r prime s 34 §13.6. Bac k t o th e siev e 34

7 7 8 9 4 5 7 7 2 4 5 8

Chapter 1 4 . Holomorphi c Modula r Form s 35 3 §14.1. Quotient s o f th e uppe r half-plan e an d modula r form s 35 3 §14.2. Eisenstei n an d Poincar e serie s 35 7 §14.3. Thet a function s 36 1 §14.4. Modula r form s associate d t o ellipti c curve s 36 3 §14.5. Heck e L-function s 36 8 §14.6. Heck e operator s an d automorphi c L- functions 37 0 §14.7. Primitiv e form s an d specia l basi s 37 2 §14.8. Twistin g modula r form s 37 6 §14.9. Estimate s fo r th e Fourie r coefficient s o f cus p form s 37 8

viii C O N T E N T

S

§14.10. Average s o f Fourie r coefficient s 38

0

Chapter 1 5 . Spectra l Theor y o f Automorphi c Form s 38 3 §15.1. Motivatio n an d geometri c preliminarie s 38 3 §15.2. Th e laplacia n o n H 38 5 §15.3. Automorphi c function s an d form s 38 6 §15.4. Th e continuou s spectru m 38 7 §15.5. Th e discret e spectru m 38 9 §15.6. Spectra l decompositio n an d automorphi c kernel s 39 1 §15.7. Th e Selber g trac e formul a 39 3 §15.8. Hyperboli c lattic e poin t problem s 39 8 §15.9. Distributio n o f lengt h o f close d geodesie s an d clas s number s 40 1 Chapter 1 6 . Sum s o f Kloosterma n Sum s 40 §16.1. Introductio n 40 §16.2. Fourie r expansio n o f Poincar e serie s 40 §16.3. Th e projectio n o f Poincar e serie s o n Maas s form s 40 §16.4. Kuznetsov' s formula s 40 §16.5. Estimate 1 s fo r th e Fourie r coefficient s 4 §16.6. Estimate s fo r sum 1 s o f Kloosterman sum s 4

3 3 4 6 6 3 5

Chapter 1 7 . Prime s1 i n Arithmeti c Progression s 4 9 §17.1. Introductio n 4 9 §17.2. Bilinea r form s i n arithmeti c progression s 42 1 §17.3. Proo f o f th e Bombieri-Vinogrado v Theore m 42 3 §17.4. Proo f o f th e Barban-Davenport-Halbersta m Theore m 42 4 Chapter 1 8 . Th e Leas t Prim e i n a n Arithmeti c Progressio n 42 §18.1. Introductio n 42 §18.2. Th e log-fre e zero-densit y theore m 42 §18.3. Th e exceptiona l zer o repulsio n 43 §18.4. Proo f o f Linnik' s Theore m 43

7 7 9 4 9

Chapter 1 9 . Th e Goldbac h Proble m 44 §19.1. Introductio n 44 §19.2. Incomplet e A-function s 44 §19.3. A ternar y additiv e proble m wit h A b 44 §19.4. Proo f o f Vinogradov' s thre e prime s theore m 44

3 3 5 6 7

Chapter 20 . Th e Circl e Metho d 44 9 §20.1. Th e partitio n numbe r 44 9 §20.2. Diophantin e equation s 45 6 §20.3. Th e circl e metho d afte r Kloosterma n 46 7 §20.4. Representation s b y quadrati c form s 47 2 §20.5. Anothe r decompositio n o f the delta-symbo l 48 1 Chapter 21 . Equidistributio n 48 §21.1. Weyl' s criterio n 48 §21.2. Selecte d equidistributio n result s 48 §21.3. Root s o f quadrati c congruence s 49 §21.4. Linea r an d bilinea r form s i n quadrati c root s 49

7 7 8 4 6

TABLE O F CONTENT S i

x

§21.5. A Poincar e serie s fo r quadrati c root s 49 8 §21.6. Estimatio n o f th e Poincar e serie s 50 1 Chapter 22 . Imaginar y Quadrati c Field s 50 3 §22.1. Binar y quadrati c form s 50 3 §22.2. Th e clas s grou p 50 8 1 §22.3. Th e clas s grou p L-function s 5 1 1 §22.4. Th e clas s numbe r problem s 5 7 §22.5. Splittin g prime s i n Q(y/D) 52 0 §22.6. Estimation s fo r derivative s L ^ ( 1 , X D ) 52 3 Chapter 23 . Effectiv e Bound s fo r th e Clas s Numbe r 52 9 §23.1. Landau' s plo t o f automorphi c L-function s 52 9 §23.2. A partitio n o f A ^ ( \ ) 53 1 §23.3. Estimatio n o f S 3 an d S 2 53 3 §23.4. Evaluatio n o f 5 i 53 4 §23.5. A n asymptoti c formul a fo r A^)(I ) 53 6 §23.6. A lowe r boun d fo r th e clas s numbe r 53 8 §23.7. Concludin g note s 54 0 §23.A Th e Gross-Zagie r L-functio n vanishe s t o orde r 3 54 1 Chapter 24 . Th e Critica l Zero s o f th e Rieman n Ze t a Functio n 54 §24.1. A lowe r boun d fo r N 0(T) 54 §24.2. A positiv e proportio n o f critica l zero s 55

7 7 0

Chapter 25 . Th e Spacin g o f th e Zero s o f th e Rieman n Ze t a-Function 56 §25.1. Introductio n 56 §25.2. Th e pai r correlatio n o f zero s 56 §25.3. Th e n-leve l correlatio n functio n fo r consecutiv e spacin g 57 §25.4. Low-lyin g zero s o f L-function s 57

3 3 4 0 2

Chapter 26 . Centra l Value s o f L-function s 57 §26.1. Introductio n 57 §26.2. Principl e o f th e proo f o f Theore m 26. 2 58 §26.3. Formula s fo r th e firs t an d th e secon d momen t 58 §26.4. Optimizin g th e mollifie r 58 §26.5. Proo f o f Theore m 26. 2 59

7 7 0 2 9 5

Bibliography 59

9

Index

611

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PREFACE This boo k show s th e scop e o f analyti c numbe r theor y bot h i n classica l an d modern directions . Ther e ar e no division lines ; in fact ou r inten t i s to demonstrate , particularly fo r newcomers , th e fascinatin g countles s interrelations . O f course , ou r picture o f analyti c numbe r theor y i s by n o mean s complete , bu t w e tried t o fram e the materia l int o a portrai t o f a reasonabl e size , ye t providin g a self-containe d presentation. We were writing thi s book i n a period o f time durin g an d afte r teachin g course s and working with graduate students in Rutgers University, Bordeau x University an d Courant Institute . W e than k thes e institution s fo r providin g condition s fo r bot h of u s t o wor k together . W e share d idea s o n wha t thi s boo k shoul d b e abou t wit h many of our colleagues, who gave us critical suggestions. Amon g them we would like to mentio n Etienn e Fouvry , Joh n Friedlander , Philipp e Miche l an d Pete r Sarnak . During a lon g proces s o f typin g an d preparatio n o f thi s boo k fo r publication , w e received stimulatin g encouragemen t an d technica l advic e fro m Serge i Gelfand , fo r all o f hi s hel p w e express ou r gratitude . Caro l Hame r helpe d t o polis h som e o f ou r English phrase s whil e he r littl e boy s trie d t o destro y th e Te X files withou t success . We thank the m al l fo r th e output . Henryk Iwanie c Emmanuel Kowalsk i 15 December, 200 3

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INTRODUCTION

Analytic Numbe r Theor y distinguishe s itsel f b y th e variet y o f tool s i t employ s to establis h results , man y o f whic h belon g t o th e mai n stream s o f arithmetic . I t is no t par t o f analysi s no r o f an y particula r disciplin e o f mathematics , howeve r i t does interac t indee d wit h variou s fields. Therefor e everybod y seem s t o vie w th e subject differently . Thi s vas t diversit y o f concept s o f analyti c numbe r theor y i s it s great attraction . Ou r desir e i n thi s boo k i s t o exhibi t th e wealt h an d prospect s of th e theory , it s charmin g theorem s an d powerfu l techniques . Howeve r i t i s no t our primar y objectiv e t o giv e proof s o f th e stronges t results , althoug h i n man y cases w e com e quit e clos e t o th e bes t possibl e ones . Rathe r w e favo r a reasonabl e balance betwee n clarity , completenes s an d generality . Th e boo k wa s conceive d with graduat e student s i n min d s o th e reade r will ofte n find tha t ou r emphasi s i s on reasonin g throughou t th e arguments . O f cours e ou r presentatio n i s subjective , and i n retrospec t ma y los e it s meaning . Certainl y w e d o no t alway s follo w th e original line s of discovery, bu t occasionall y w e do draw brie f historica l perspectives . Leonard Eule r mus t ge t credi t fo r th e first us e o f analytica l argument s fo r th e purpose o f studyin g propertie s o f integers , specificall y b y constructin g generatin g power series . Euler' s proo f o f th e infinit y o f prim e number s make s us e o f th e divergence o f th e zet a functio n an d th e correspondin g produc t ove r primes , whic h is name d afte r him . Thi s wa s th e beginnin g o f analyti c numbe r theory . Nex t came P . G . L . Dirichle t whos e creatio n o f the theor y o f L- functions fo r characters , resulting i n th e proo f o f th e infinit y o f prime s i n arithmeti c progressions , make s him th e tru e fathe r o f analyti c numbe r theory . Pro m thes e earl y day s t o moder n times th e distributio n o f prim e number s constitute s th e cor e o f th e subject . Thi s will b e apparen t i n th e cours e o f ou r book . Th e first tw o chapter s cove r question s of prime s u p t o th e elementar y method s o f P . Tchebychev . Chapter 3 provides definition s an d basi c propertie s o f Dirichle t character s an d the Gaus s sums . Character s o n ideal s o f imaginar y quadrati c fields ar e als o intro duced, no t onl y becaus e the y pla y a supportin g rol e i n subsequen t chapter s bu t t o show a bi t o f analytic numbe r theor y beyon d th e traditiona l domai n o f the rationa l integers a s well ; ther e wil l b e othe r example s throughou t th e book , fo r instance , elliptic curves . Poisson summatio n fo r numbe r theor y i s wha t a ca r i s fo r peopl e i n moder n communities - i t transport s thing s t o othe r place s an d i t take s yo u bac k hom e when applie d nex t tim e - on e canno t liv e without it . Chapte r 4 presents a classica l account o f this basic technology. Man y reader s do realize now, others will figure out later, tha t w e ar e alread y talkin g abou t idea s o f modula r forms . Bu t w e continu e our consideration s alon g traditiona l line s (bot h classica l an d mor e recen t ones ) before th e concep t o f modularit y take s th e leadin g position . 1

2

INTRODUCTION

The celebrate d memoi r o f B . Rieman n o n th e zet a functio n i s embedde d i n the contex t o f abstrac t L-function s i n Chapte r 5 . I t i s no t ou r styl e t o conside r things i n term s mor e genera l tha n necessary , s o definin g a clas s o f L-function s which suits minimum requirement s o f our forthcomin g application s wa s not withou t difficulty an d hesitation . I n thi s wa y we could conve y to dedicate d researcher s tha t generalizations ar e no t alway s straightforward . Fo r instance , t o establis h th e zero free region for L-functions o f degree > 1 one cannot rel y on the same principles as for the Dirichlet L-functions . Th e key ingredient i s the Rankin-Selberg convolution. O n the othe r hand , th e proble m o f exceptiona l zer o i s resolve d fo r man y automorphi c L-functions o f degree > 1 (not withou t cleve r constructions ) whil e i t remain s ope n for th e L-function s wit h rea l characters . Furthermor e i n Chapte r 5 a messag e i s sent tha t a bette r lif e exist s i n th e worl d o f automorphi c form s tha n i n th e zo o of degree on e L-functions . Analytic number theor y does not mea n non-elementary. Th e first autho r recall s that hi s first seriou s encounte r wit h analyti c numbe r theor y starte d b y readin g th e lovely boo k o f A.O . Gelfon d an d Yu.V . Linnik , "Elementar y Method s o f Analyti c Number Theory" . Whe n a n ambitiou s beginne r start s fro m ther e her/hi s lov e o f the subjec t i s seale d forever . Tr y i t yourself ! On e i s instantl y capture d b y siev e methods. I n thi s boo k w e do no t hav e spac e t o giv e justice t o thi s marvelou s idea , nevertheless Chapte r 6 should suffic e fo r basi c applications . Next come s the "Larg e Sieve" , which is not a sieve but a name fo r othe r things . Yes, i t di d originat e fro m a shor t pape r b y Linni k o n a siev e problem , bu t i t too k time t o recogniz e th e tru e natur e o f thes e ideas . I n Chapte r 7 we revea l ou r view point an d th e crucia l attribute s (spectra l completeness , orthogonality) , the n w e demonstrate th e amazin g powe r o f th e larg e siev e o n selecte d ol d an d ne w prob lems. Othe r feature s o f the large sieve are scrutinized showin g the good an d th e ba d sides. Fo r example , th e approac h usin g th e dualit y principl e i s fruitful fo r harmon ics o f degre e on e (characters ) whil e producin g poo r result s fo r harmonic s o f large r degree (lik e fo r exampl e th e eigenvalue s o f Heck e operators). Th e controvers y ove r the prope r plac e o f th e larg e siev e i s academic . Simpl y speakin g th e larg e siev e inequalities ar e part s o f bilinea r form s theory . Estimates fo r exponentia l sum s ar e th e first tool s whic h deepl y penetrat e th e problems o f analyti c numbe r theor y beyon d natura l structures . Thes e canno t b e grasped b y harmoni c analysi s alone . Se e what cleve r us e peopl e mad e o f th e prop erty tha t a shifte d interva l i s anothe r interval , tha t addin g a n intege r t o a se t o f integers yield s agai n a se t o f integer s (sorry , prime s ar e no t preserved!) . W e chal lenge algebraist s t o fin d a structura l explanatio n o f th e powe r o f suc h arguments ! They shoul d rea d Chapte r 8 t o fin d wha t H . Wey l buil t ou t o f thes e observa tions. Va n de r Corpu t an d Vinogrado v ar e als o th e mai n figures fro m th e earl y stages o f tha t discipline . A lo t o f wor k an d talkin g wen t int o ou r presentatio n o f Vinogradov's method , becaus e i t i s not quit e correctl y explaine d i n numerou s pub lications. A t som e poin t Vinogrado v depart s fro m th e Wey l differencin g proces s and treat s multi-dimensiona l exponentia l sum s a s bilinea r form s (thi s i s th e wa y we think o f i t anyway) . The nex t tw o chapters sho w more recent technolog y which was developed t o replace the unprove n Rieman n hypothesi s i n applications t o th e distributio n o f prime numbers. W e ar e talkin g abou t estimate s fo r th e numbe r o f zero s o f L-function s i n vertical strip s whic h ar e positivel y distance d fro m th e critica l line . Hopefull y i n a future on e wil l sa y w e wer e wastin g tim e o n studyin g th e empt y set . Grea t idea s

INTRODUCTION

3

are camouflage d ther e i n argument s o f enormou s complexity , s o thi s migh t no t b e enjoyable fo r everyon e a t first . Howeve r i f you thin k th e Rieman n hypothesi s i s not provable i n you r lifetime , pleas e rea d an d admir e thes e unconditiona l substitutes . Special mentio n goe s t o Hug h Montgomery , Marti n Huxle y an d Matt i Jutil a fo r the mos t origina l contributions . Although w e are primar y intereste d i n rationa l integer s on e ca n lear n an d ben efit a lo t fro m arithmeti c o f other fields . No t onl y fro m th e numbe r field s o r p-adi c fields, bu t indirectl y from th e fields of finite characteristi c a s well. Particularl y fruit ful ar e th e method s o f exponentia l sum s ove r th e finit e fields . I n Chapte r 1 1 we prove (amon g othe r things ) th e Rieman n hypothesi s fo r specia l curve s whic h yield s the celebrated estimat e of Weil for Kloosterma n sums . Th e Kloosterma n sum s hav e been employe d t o solve various problems o f analytic numbe r theor y fro m th e begin ning o f thei r creatio n i n 1 926 . W e als o mentio n briefl y th e stat e o f th e knowledg e of exponentia l an d characte r sum s ove r algebrai c varieties . Application s o f thes e are harde r t o make , ye t ther e i s a handfu l o f example s i n th e literature . I t wa s a painfu l decisio n t o exclud e al l bu t th e simples t fro m presentatio n i n thi s book . Otherwise t o d o ful l justic e fo r thes e highl y sophisticate d idea s w e woul d hav e t o choose th e mos t complicate d applicatio n fo r whic h w e hav e n o room . I t suffice s t o say that a preparation o f a given problem o f analytic numbe r theor y t o a n estimat e for characte r su m ove r varietie s ca n b e th e state-of-the-ar t i n it s ow n right , neve r mind tha t th e final argumen t i s powered by the outside forces o f algebraic geometry . Dirichlet character s ar e alread y discusse d i n Chapte r 3 and w e return t o the m in Chapte r 1 2 to trea t ver y shor t characte r sums . Agai n on e mus t b e inventiv e t o break limit s o f natura l structures . Burges s theore m i s a fin e example . Sums over primes ar e treated i n the next chapter . Whe n Vinogrado v succeede d in estimating sums over primes of additive characters, which he needed for a solution to th e ternar y Goldbac h proble m i n conjunctio n wit h th e circl e method , i t wa s a shocking result . Befor e hi m th e Gran d Rieman n Hypothesi s coul d d o th e job , bu t keep in mind that th e Riemann hypothesi s is still not established . Th e original ideas of Vinogradov were borrowed fro m combinatoria l sieve and were rather complicated . Recently develope d identitie s offe r muc h simple r treatment s o f mor e genera l sum s over primes . A s the y shar e th e sam e fundamenta l principl e (reducin g th e su m t o bilinear forms ) th e result s ar e prett y muc h th e same , s o th e choic e o f th e metho d is a matte r o f tast e an d technica l convenience . T o captur e th e ke y element s i n Chapter 1 3 we develop mor e tha n on e identity . A popular criterio n for analyti c number theor y is that comple x variable analysi s is bein g used . Perhap s i t i s bette r t o sa y harmoni c analysis , sinc e th e actio n o f the latte r i s mor e profound . Fo r a lon g time , analyti c numbe r theor y flourishe d exclusively from abelia n harmonic analysis, that i s to say from th e Fourier transfor m in IR n. Ther e i s stil l a grea t potentia l i n thi s classica l analysi s t o b e explored . However muc h stronger fertilizer s bega n to act o n the soi l of analytic number theor y in recen t times . Thes e ar e automorphi c functions . O f course , modula r form s hav e been driving algebraic aspect s of number theor y muc h longer , bu t i n a limited scop e (confined t o holomorphic forms) . Ne w resources of automorphic theor y are found i n the spectra l analysis , th e foundatio n o f which wa s le d b y H . Maas s an d A . Selber g at th e turn o f the 1 940' s (real-analyti c cus p forms, Eisenstei n series , trace formula) . In simpl e term s a non-abelia n harmoni c analysi s foun d it s rol e i n analyti c numbe r theory. Trul y effectiv e expansio n o f spectra l method s int o analyti c numbe r theor y began abou t twent y fiv e year s ago , changin g th e fac e o f eithe r subjec t irrevocably .

4

INTRODUCTION

This boo k barel y addresse s th e fascinatin g issue s o f th e ne w directio n throughou t Chapters 1 4 , 1 5 , an d 1 6 . Ou r feature d applicatio n i s t o estimatio n fo r sum s o f Kloosterman sums . Thi s i s a goo d choic e (i f n o mor e ca n b e accommodated) , because th e reade r ca n appreciat e th e ne w tool s b y comparin g wit h th e earlie r results derive d i n Chapte r 1 1 b y algebrai c considerations . Anothe r applicatio n o f the spectra l theor y o f automorphi c form s t o a n arithmetica l questio n i s presente d in Chapte r 21 , tha t i s t o th e equidistributio n o f root s o f quadrati c congruence s of prim e moduli . Th e spectra l theor y continue s t o gro w extensively , s o i t woul d be prematur e t o wra p i t u p her e o r i n an y othe r book . Fo r furthe r readin g w e recommend [1 3] , [Sa3] . Although th e spectra l method s o f automorphi c form s predominat e curren t re search i n analyti c numbe r theory , th e traditiona l problem s continu e gettin g ou r attention wit h respectfu l intensit y throughou t th e remainin g chapters . Grea t trea sures o f th e subjec t mustn' t b e burie d i n th e past . Firs t o f al l a newcome r shoul d learn th e storie s o f prime s i n arithmeti c progression s t o larg e moduli . I n Chapte r 17 she/h e wil l find ho w E . Bombier i an d A.I . Vinogrado v bypasse d th e Rieman n hypothesis t o establis h (b y th e larg e siev e an d othe r means ) unconditiona l result s with application s a s goo d a s on e ca n ge t fro m th e R H itself . O f cours e ou r argu ments ar e no t identica l wit h th e origina l one s (o f 1 965 ) sinc e w e take advantag e o f later simplification , i n grea t measur e du e t o P.X . Gallagher . Chapter 1 8 goes further bac k to 1 94 4 when Linnik gave an extraordinary boun d for th e leas t prim e i n a n arithmeti c progression . Fo r a lon g tim e thi s boun d wa s considered a s th e mos t difficul t theore m i n analyti c numbe r theory . An d yes , i t i s still har d b y today' s standards , an d on e ca n stil l lear n a lo t fro m th e technolog y applied! Face d with the obstacle of the exceptional zero, Linnik brings the repulsio n effect (h e calls it Deuring-Heilbronn phenomenon ) t o a new level; amazingly enoug h he turn s th e proble m t o hi s advantage ! Thi s i s a fascinatin g developmen t i n th e history o f analyti c numbe r theor y whic h w e recommen d on e shoul d maste r fo r a better understandin g o f the statu s o f th e exceptiona l zer o today . Once upo n a tim e th e famou s Goldbac h proble m wa s wort h a millio n dolla r prize award . Fo r application s th e proble m (representation s o f eve n integer s b y th e sums o f two primes ) ha s n o grea t merit , bu t a s a n intellectua l challeng e on e woul d be prou d t o crac k it . Probabl y somethin g ne w abou t prim e number s woul d b e revealed then . Rea d Chapte r 1 9 to improv e you r chances . Chapter 2 0 i s serious . Her e analyti c method s stor m th e domai n o f diophan tine equations , whic h fro m ancien t Greek s wa s exclusively a busines s o f arithmetic . Started b y Hard y - Ramanujan , continue d b y Hard y - Littlewoo d an d develope d substantially furthe r b y Kloosterman, th e circl e method use s orthogonality o f addi tive characters to detect equations , no t onl y to solve algebraic equation s bu t a larg e class of additive problem s ove r special integers a s well. Th e toughes t ar e the binar y additive problems. The y are not completely solved by the Kloosterman method, bu t at leas t w e get a very reliable picture o f what th e true asymptoti c fo r th e numbe r o f solutions shoul d be . Kloosterma n sum s whic h w e covered i n the precedin g chapter s are instrumenta l i n th e circl e methods . After classica l idea s w e propos e a mor e direct varian t whic h i n principl e shoul d produc e th e sam e results , howeve r withou t employing Kloosterma n sums . On e shoul d rea d Chapte r 2 0 wit h a n ope n mind , separate technica l (stil l attractive ) element s fro m conceptua l device s t o see clearl y

INTRODUCTION

5

the connection s wit h modular forms . Certainl y Kloosterma n an d Rademache r wer e aware o f thes e intrinsi c connections , whil e the y ar e overlooke d b y som e specialist s in th e circl e method . Equidistribution problem s fo r sequence s o f specia l integers , lattic e point s i n various domains , solutio n t o diophantin e equations , etc , constitut e a heav y indus try over the analyti c number theory. W e regret ther e is no space to run this industr y in full capacit y i n the book . Th e boo k of M.N. Huxle y [Hu4 ] treat s onl y the lattic e point problems , howeve r quit e deeply . I n Chapte r 2 1 we are dealin g wit h th e prob lem o f distributio n o f roots o f a quadrati c equatio n reduce d modul o prime . A s th e prime modulu s tend s t o infinit y w e sho w tha t th e root s ar e uniforml y distributed . The argument s includ e almos t everythin g tha t w e develope d i n th e boo k s o far , thus showin g tha t th e industr y i s robust . Because o f failur e o f th e uniqu e factorizatio n o f algebrai c integers , th e arith metic of number fields is not a s easy as for rationa l numbers an d sometime s perplex ing. Th e complexity i s measured b y the order o f the ideal class group. Naturall y th e case of imaginary quadrati c fields receive d th e first an d th e mos t attentio n becaus e units do not interfere . W e do know that th e class number grow s to infinity (s o there is only a finite numbe r o f imaginary quadrati c fields with a fixed class number), bu t the seriou s issu e i s t o estimat e th e clas s numbe r effectively . Chapte r 2 2 describe s the proble m thoroughl y an d prepare s th e groun d fo r th e advance s i n Chapte r 23 . The effective lowe r bound fo r the class number (du e to D. Goldfeld) ma y not appea r strong fo r demandin g researchers , ye t i t i s deep wit h respec t t o result s take n fro m other sources . Firs t o f all it use s the Gross-Zagie r formul a fo r //-function s o f ellipti c curves a t th e centra l point . W e d o provid e a substantia l overvie w o f th e involve d arguments fro m ellipti c curves , althoug h thes e ar e mor e geometri c tha n analytic . The analyti c argument s themselve s ar e quit e delicate . Actuall y the y cam e first, the L-function s o f ellipti c curve s bein g supplementary . Indee d w e worke d ou t a n effective lowe r boun d fo r th e clas s number whic h depend s o n th e orde r o f vanishin g of general L-function s o f degre e two , suspectin g tha t th e requirement s ar e satisfie d by quit e a fe w o f them . In Chapte r 2 4 we prove a very classical result o f Selberg that a positive propor tion o f zero s o f th e Rieman n zet a functio n lie s o n th e critica l line . Thi s i s a goo d place t o lear n abou t th e mollificatio n technique s ( a kin d o f smoothing) , whic h i s used i n man y work s toda y an d wil l reappea r i n Chapte r 26 . Assuming th e Rieman n hypothesis , H.L . Montgomery reveale d i n 1 97 4 that th e distribution o f zero s o f £(s ) follow s th e behavio r o f eigenvalue s o f certai n "ensem bles" o f unitar y matrices . Mor e recentl y physicist s joine d th e tea m o f worker s i n number theory , creatin g a ne w excitemen t an d hop e fo r finding a pat h t o a proo f of the Rieman n hypothesis . Thi s i s the mai n objectiv e o f the so-calle d rando m ma trix theory , on e o f th e mos t popula r subjec t an d drivin g force s o f curren t analyti c number theory . I t offer s reliabl e model s fo r predictin g th e behavio r o f arithmetica l quantities whic h fo r a lon g tim e wer e shroude d i n mystery . Th e consistenc y o f the rando m matri x theor y wit h th e harmon y o f integer s stil l seem s quit e surpris ing. Whateve r th e futur e o f this enterpris e wil l be , du e t o th e curren t cooperation , analysis i s closer t o arithmeti c tha n eve r before . A subject o f suc h magnitud e can not b e full y presente d i n a shor t space . Therefor e i n Chapte r 2 5 w e stic k t o th e original them e o f th e correlatio n o f zero s o f £(s ) an d it s variation s o n th e zero s o f

6

INTRODUCTION

families o f automorphi c L-function s whic h ar e nea r th e centra l point . W e leave i t for th e reader t o judge whethe r th e idea s o f random matri x theor y ar e realistic fo r launching a n attac k o n the Rieman n Hypothesis . In recen t investigation s th e centra l value s o f L-function s appea r i n a variet y of formula s wit h vanishin g o r non-vanishin g assumptions . Tak e fo r exampl e [IS2 ] where a n effectiv e lowe r boun d fo r th e clas s numbe r o f imaginar y quadrati c fields is derive d essentiall y fro m th e non-vanishin g o f centra l value s o f familie s o f L functions, t o th e contrar y o f the vanishin g requirement s i n the previou s investiga tions. Anothe r exampl e i s the formul a o f T. Watso n [Wa ] by mean s o f which th e quantum-ergodicity conjectur e (tha t i s the equidistributio n o f Maas s cus p forms ) is reduced t o a subconvexity boun d fo r certai n L-function s o f degree four . W e consider i n detai l on e non-vanishin g statemen t whic h ha s application s t o arithmeti c geometry. We hope thi s boo k will show the picture o f analytic numbe r theor y i n plenty of colors. Howeve r we must sa y that a lot of significant topic s are left out . Missin g are the dispersio n method , th e amplificatio n metho d (se e [M2]) , som e analyti c tech niques fro m diophantin e approximation s an d transcendence . Moreove r probabilit y arguments ar e barel y exposed , an d w e didn' t touc h ergodi c theor y either , whos e impact o n number theor y ha s been fel t strongl y i n the last years . We als o try to show som e detail s o f the powerful theorie s whic h ar e developin g as th e mos t usefu l ne w tools fo r analyti c numbe r theory , i n particula r th e theor y of higher-degree automorphi c form s an d their L-functions , an d algebraic geometry ; young researcher s i n particular shoul d b e encouraged t o develop expertis e i n thes e subjects. I t i s certain tha t spectacula r application s hav e onl y begu n an d more wil l be ope n t o thos e wh o understan d bot h sides . Dually , arithmeti c geometr y an d algebraic numbe r theor y als o giv e an d promis e a wealt h o f ne w questions, o r new aspects o f old ones, wher e th e skills an d technique s o f analytic numbe r theor y wil l be tested to the utmost. Hopefull y the y will bring rich rewards to those who will try to com e to these ope n fields... W e barely mentio n som e questions relate d t o ellipti c curves bu t w e believe tha t ther e i s muc h mor e t o discover . Th e dee p conjecture s of Lan g an d Trotte r [LT ] ar e alread y quit e popular , an d a fe w othe r challengin g problems ma y be foun d i n [Kol] . The exercise s insid e eac h sectio n serv e a dua l purpose , som e ar e t o improv e the reader' s skill , th e other s serv e a s additiona l informatio n abou t th e subject . Historical remark s ar e brief , t o giv e som e orientatio n i n th e developmen t o f th e matter, rathe r tha n t o credi t exhaustivel y th e inventors . Th e only advic e w e offe r to new researchers is read! read ! read ! man y papers with complete proofs. Knowin g a resul t i n analytic numbe r theor y i s only the first ste p to liking it; more importan t and rewardin g i s to understan d th e argument s o f its proof . Ou r viewpoin t i s tha t making mathematic s shoul d no t b e rate d lik e breakin g spor t records . Sometime s the stronges t resul t i s boring whil e a slightl y weake r on e generates grea t pleasure . Formal prerequisite s fo r muc h o f the boo k ar e rather slight , no t goin g beyon d differential calculus , comple x analysi s an d integration, especiall y Fourie r serie s and integrals. I t i s more importan t fo r the reader t o have or acquire a good understand ing o f how to manipulat e inequalitie s an d not simpl e identities .

INTRODUCTION

7

In late r chapter s automorphi c form s becom e important . W e have include d tw o survey chapters, ye t w e expect tha t man y readers will have already som e knowledg e of this importan t topic , o r wil l stud y i t independently . In som e section s (fo r instanc e Section s 5.1 3 an d 5.1 4) , whic h ar e intende d a s convenient reference s fo r certai n fact s an d result s which ar e hard t o locate in prope r form i n th e literature , w e assum e tha t th e reade r ha s som e familiarit y wit h othe r topics, suc h a s representation s o f group s an d algebrai c geometry . Sections o f this boo k wer e writte n ove r a perio d o f time , therefor e reader s wil l notice a sligh t chang e o f styl e an d repetition . W e thin k tha t a smal l redundanc y is helpfu l fo r readin g lon g arguments . Occasionall y th e sam e objec t i s introduce d again i n a differen t chapte r i n loca l terminolog y whic h shoul d b e mor e familia r i n a particula r context . W e believ e thi s flexibilit y i s justified fo r comfort , eve n a t th e expense o f losin g uniqueness . Our notation s ar e mostl y standard . Bu t sinc e inequalitie s wit h unspecifie d constants ar e the lifeblood o f analytic number theory , an d sinc e there ar e sometime s controversies o n thi s subject , w e spel l ou t th e meanin g o f th e variou s compariso n symbols 0 ( ) , o() , ~ , x o r C i s bette r see n a s th e sequenc e A = (a n ) wit h a n = f(n). Ver y ofte n i n suc h contex t A i s supporte d o n number s of primar y interest , an d a n i s jus t a multiplicity , o r som e sor t o f weight , whic h is introduce d whe n countin g suc h numbers . A differen t jo b i s assigne d t o certai n functions / : N — > C (suc h a s Dirichle t character s o r Heck e eigenvalues ) whic h we cal l "arithmeti c harmonics. " Thes e pla y a n instrumenta l rol e i n analyzin g th e primary sequenc e A = (a n). Essentiall y th e arithmeti c harmonic s ar e employe d with th e primar y sequenc e A = (a n) t o produc e a famil y o f twiste d sequence s Af = (a nf(n)). Th e twiste d sequence s wit h appropriat e harmonic s ar e capabl e o f selecting a special subsequence of A whic h is our target (thin k of Dirichlet character s being employe d t o detec t prime s i n arithmeti c progressions) . Thanks t o th e additiv e an d multiplicativ e structure s o f integers th e tw o impor tant classe s o f arithmeti c function s ar e distinguished . A functio n / : N— > C i s a n additive functio n i f i t satisfie s (1.1) f(mn)

= f(m) + f(n)

for m,n relativel y prime . I f thi s propert y hold s fo r al l ra, n , the n / i s sai d t o b e completely additive . Fo r exampl e f{n) = log n i s completel y additive . Similarly , a 9

1. ARITHMETI C FUNCTION S

10

function / : N—» C i s multiplicative i f i t satisfie s (1.2) f(mn)

= f(m)f(n)

for ra,n relativel y prime , an d / i s completel y multiplicativ e i f (1 .2 ) hold s fo r al l ra,n. Fo r exampl e f(n) = n~ s wit h 5 G C i s completel y multiplicative . Obviously , the additiv e an d th e multiplicativ e function s ar e determine d b y thei r value s a t prime powers . W e hav e / ( l ) = 0 i f / i s additive , / ( l ) = l i f / i s multiplicativ e no t identically zero . 1.2. Generatin g series . To an arithmeti c functio n / w e shall attac h th e tw o most natura l infinit e serie s

(1.3) E

f(z)

= £/(n)z" , n

(1.4) D

f(s)

^/(n)n" s,

=

n

where z, s ar e comple x variable s (i n th e cas e o f th e powe r serie s Ef(z) w e ma y also includ e n = 0 i f /(0 ) i s defined) . Thes e ar e calle d th e generatin g series , o r functions, o f / . Othe r kind s o f generatin g function s ca n b e attractiv e a s wel l t o capture distinc t propertie s o f / . Fo r man y arithmeti c function s th e correspondin g generating serie s converge s absolutel y i n a smal l domain , an d i t i s a n interestin g question ho w fa r th e serie s ca n b e analyticall y continued ? I f th e generatin g se ries extend s beyon d th e rang e o f absolut e convergence , the n thi s propert y usuall y manifests som e grou p structur e i n th e coefficient s f(n) ("random " serie s canno t be continued) . Fo r exampl e th e analyti c continuatio n o f Arti n L- functions i s inti mately relate d t o th e reciprocit y law s i n numbe r fields (i n th e abelia n cas e a t an y rate), th e analyti c continuatio n o f Hasse-Wei l L- functions i s a majo r ste p toward s the modularit y o f th e correspondin g ellipti c curve s (ove r Q) , an d s o on . The generating power series were introduced b y L. Euler (1 707-1 783 ) fo r study ing specia l additiv e problem s (whe n doin g s o h e wa s th e first t o mi x analysi s wit h arithmetic). Le t Ef(z) an d E g(z) b e th e serie s fo r / an d g. The n th e produc t (1.5) E

f(z)Eg(z)

=

J2 Hn)z

n

n

yields th e powe r serie s fo r th e functio n h give n b y (th e additiv e convolution )

(1.6) h(n)

= J2 fW9(™)£-\-m=n

Applying Cauchy' s theore m on e expresse s th e coefficien t h(n) b y th e contou r inte gral (1.7) h(n)

= ^- f

E

1 n f(z)Eg(z)z' '

dz.

J\z\=r

Hence, give n reasonabl e analyti c propertie s o f Ef(z)E g(z) on e ca n deduc e a goo d estimate fo r h(n), o r even a n asymptoti c formul a a s n tend s t o oo. O f course, Eule r did no t kno w Cauchy' s theorem , s o hi s idea s wer e limite d t o th e powe r serie s fo r which on e has a secondary expression , givin g a n exac t formul a fo r f{n) rathe r tha n

1. A R I T H M E T I C F U N C T I O N

11

an approximatio n (stil l no t a tautology). W e give three cut e examples . Firs t i s th e identity oo

oo

$>" = II(1 + *2m) = ( 1 -*r 1 1 0

which i s nothin g bu t a n analyti c statemen t o f th e uniquenes s o f binar y expansio n of natura l numbers . Her e i s anothe r identity : oo

oo

n

X>(n)* = II( 1 -* ro )~ 1 1 0

where p(ri) i s the partition function . Les s obvious is the following formul a o f Jacobi: oo

(1.8) £ V

oo

* = JJ(1 - z m){\ - z m+1 /2f.

1 — oo

About ninet y year s ag o th e integra l representatio n (1 .7 ) fo r th e additiv e equatio n (1.6) gav e birth t o th e circl e method. Thes e idea s wil l be develope d i n Chapte r 20 . By changin g th e variabl e z int o (1.9) e(z)

= e

2niz

the powe r serie s (1 .3 ) i s see n a s a Fourie r series , whic h i s a commo n practic e i n modern analyti c numbe r theory . The generatin g serie s Df(s) i s calle d th e Dirichle t serie s afte r bein g use d i n Dirichlet's fundamenta l work s o n prime s i n arithmeti c progressions , thoug h th e special cas e oo

S

(1.10) C(*)=£n1

was alread y considere d b y Euler . The Dirichle t serie s i s particularl y attractiv e fo r multiplicativ e function s (bu t not limite d t o thi s case) . For , i f / i s multiplicative, the n (l.n) D

f(s)

=

n(i+

f(p)p~s+f(p2)p~2s+•••)

p

provided th e involve d serie s i n prim e power s an d th e produc t ove r prime s (calle d the Eule r product ) converg e absolutely . I n particular , w e hav e

(i.i2) C(*

) = II(i-p-T 1 V

if R e (s) > 1 . Thi s representatio n o f £(s ) a s a n Eule r produc t i s a n expressio n i n analytic languag e o f the uniqu e factorization o f natural number s int o distinct prim e powers.

1. ARITHMETI C FUNCTION S

12

1.3. Dirichle t convolution . Assuming tha t th e Dirichle t serie s Df(s),D g(s) converg e absolutel y i t follow s that th e product Df(s)D g(s) i s also give n b y a Dirichle t series , namel y w e have oo

Df(s)Dg(s) =

s

J2 Kn)n~ 1

with th e coefficient s

(1.13) Mn

) = £/(d)ffQ) d\n

The arithmeti c functio n h defined b y (1 .1 3 ) (neve r min d th e convergenc e o f gener ating series ) i s called th e Dirichle t (o r multiplicative) convolutio n o f / an d g, an d it i s denoted b y f *g. The se t of all arithmetic function s wit h th e usual additio n + an d the operatio n • i s a commutative ring . Th e function 5 : N -> C whose Dirichlet serie s is Ds(s) = 1 is the uni t elemen t o f this ring , i.e.,

J 1 i f n= 1 ,

(1.14) S(n)

~ \ 0 ifn>l

.

The functio n ((s) define d i n R e (s) > 1 by (1 .1 0 ) i s calle d th e Rieman n zet a function (afte r hi s semina l memoi r [Rie]) , it s Dirichle t coefficient s mak e th e con stant functio n l(n ) = 1 for al l n ^ 1 . B y virtu e o f (1 .1 2 ) th e invers e o f £(s ) ha s also a Dirichle t serie s expansion , namel y (1.15) '

-TT 1 ^ \_^M» 11V psj ^m

C(s)

s

0 '

P

say, wit h coefficient s (1.16) n(m)

= |

(-l)r 0

if m = pi . .. pr wit h pi ,.. . , p r distinc t otherwise.

The functio n fi(m) wa s introduced i n 1 83 2 by A. F. Mobius , an d it bear s hi s nam e ever since . Moreover , b y th e Eule r produc t (1 .1 2 ) th e logarith m o f ((s) ha s th e Dirichlet serie s expansio n

(i.i7) iogC0

0= £ £ r y

•is

e=i P

Recall tha t th e Dirichle t convolutio n * correspond s t o th e multiplicatio n o f th e generating Dirichle t series , therefor e th e identit y £(s ) • (~ 1 (s) = 1 reads a s

(1.18) a(m

) = 5>(l

.

1. A R I T H M E T I C F U N C T I O N

13

MOBIUS INVERSION . For any / , g : N — > C the following two relations are

equivalent:

(1-19) g(n)

= J2f(d), d\n

(1.20) /(»

) = 5>(d)sQ)d\n

REMARK. T O be accurate wit h th e history th e inversion formula s i n the above form wer e state d onl y i n 1 85 7 by R . Dedekind . Th e origina l versio n state d b y Mobius wa s somewhat different , i t amount s t o sayin g tha t fo r an y rea l variabl e functions F , G : [1, x]— > C the following tw o relations ar e equivalent:

= Y. F

(1.21) G{x)

(xh),

n^x

(1.22) F(x)

= ^2 v(rn)G(x/m). m^x

The Mobiu s functio n i s multiplicative . I f / , g ar e multiplicative, the n s o are / • g and / • g. I f g is multiplicative, the n d

(1.23) E ^

) ^ d ) = ll(1-5(P))-

d\n p\n

This produc t admit s a probabilisti c interpretation . Viewin g g(p) as a probabilit y of som e independen t event s whic h ma y occur a t p on e can think o f (1 .23 ) a s the probability tha t non e o f the events associate d wit h prim e divisor s o f n occur . 1.4. Examples . Now w e give a larg e sampl e o f arithmeti c function s whic h on e encounters i n analytic numbe r theory . Man y othe r function s wil l be introduced i n due course. The diviso r functio n r(n) i s the number o f positive divisor s o f n, so we have oo

(1.24) e(s)

=

y

£r(n)n-°. 1

More generall y Tk(n) denote s th e number o f representations o f n a s the product of k natura l numbers , s o its Dirichlet serie s i s ( k(s). Explicitl y

(1-25) ^)=(

1 ai

t-l )--(art-ll) tt'-tf-rf'-

For an y v £ C we define o v{n) b y oo

s

(1.26) C ( s ) C ( s - ^ ) = 5 > » n 1

so it is the sum of powers o f divisors (1.27) n ) = 1-

1 . Fo r an y intege r k ^ 0 an d a rea l numbe r x > 0 , defin e

(1.47) A

fc(n,aO

= $>(d)(log^)* . d\n

Note tha t A k(n,x) depend s onl y o n th e squarefre e kerne l o f n. Prov e tha t 'k\ A A ,_ nN )/ , l_0x\g

Ak(n,x) = J2 ( ! ) ^ (

k

D

~i

Then, usin g A . < LJ~ lAk, deriv e tha t fo r n ^ x (1.48) A

fc(n,

x) f ^ J


(n)f(ri) an d A/(n ) = A(n)/(n) . There i s a variet y o f trul y interestin g arithmeti c functions . Th e theor y o f modular form s i s a basi c sourc e fo r multiplicativ e functions . A s a simpl e mode l w e pick up the functio n r(n) whic h is the number o f solutions t o a 2 + b 2 — n i n integer s a, b. Th e generatin g Dirichle t serie s fo r r(n) i s equa l t o 4(#(s) , wher e

C*(s) = J2(Na)-° a

is the zet a functio n o f th e imaginar y quadrati c field K = Q ( \ / ~ l ) whil e th e facto r 4 i s th e numbe r o f unit s i n K. Her e a run s ove r non-zer o integra l ideal s o f K. Al l these ar e principa l ideal s generate d b y th e Gaussia n integer s a = a + bi G Z[>/—T] , a ^ O , an d i f a = (a) , the n Na = a 2 + b 2. Hence , indee d 4

Cx(s) = ^ r ( n ) n -

s

.

Clearly \r{n) i s multiplicative . Th e prim e number s whic h ar e represente d a s th e sum o f tw o square s ar e characterize d i n a beautifu l theore m o f Fermat . I t i s eas y to se e tha t n o prim e p = — l(mod 4 ) ca n b e s o writte n an d Ferma t prove d tha t al l other prime s ca n be (uniquel y u p to the sig n an d orde r o f a, b). I n modern languag e this theorem o f Fermat i s just th e factorization la w in the Gaussia n domai n Z[\/—l] ; it assert s tha t p — 2 is squar e o f a prim e ideal , p = l(mo d 4 ) split s int o a produc t of tw o distinc t (comple x conjugate ) prim e ideals , p = — l(mod 4 ) remain s prime . Using th e factorizatio n la w on e ca n sho w b y verificatio n o f loca l factor s tha t

where % 4 is th e non-trivia l characte r t o modulu s 4 (w e hav e X4( n) = si n ^) an d L(s, X4 ) i s the associate d Dirichle t L- function L{S,XA)

= ^2x4(n)n~

Hence (1.51) r(n)

= 4^

X4

d\n

(d).

18

1. A R I T H M E T I

C FUNCTION S

Besides th e Dirichle t serie s there i s great interes t i n studying th e Fourie r serie s for r(n) du e t o th e additiv e natur e o f th e equatio n a 2 + b2 = n. W e hav e oo

^2r(n)e(nz) = o

0 2(z)

(recall (1 .9 ) tha t e(z) = e27rlz), wher e 0(z) i s the thet a functio n oo

(1.52) 6{z)

= ^e{n

2

z).

— OO

This serie s converge s absolutel y fo r I m (z) > 0. Mor e generally , lettin g rk(n) b e the numbe r o f representation s o f n a s th e su m o f k square s w e hav e oo

J2rk(n)e(nz) = o

0 k(z).

On one hand, th e theta function 0(z) i s recognized in analytic number theor y fo r its abilit y t o selec t squares . O n th e othe r hand , 0{z) i s usabl e du e t o th e followin g transformation rule : (1.53) ef^±l\

=v( \ (sZ I

Cld)(cz

+ d)h(z)

(J/ J

which hold s fo r an y z with I m (z) > 0 an d an y integer s a, 6, c, d wit h ad — be = 1, c = 0 (mod 4) , wher e i/(c,d) depend s onl y o n c , d. W e hav e v 2(c,d) = x±{d), so z/(c, d) takes onl y fou r value s ± 1 , ±z (se e (3.42 ) fo r th e exac t description) . I n othe r words, 0(z) i s a modular for m o f weight \ wit h multiplie r v(c,d) whil e 0 2(z) i s a modular for m o f weight on e an d characte r % 4 on r 0 (4) . I n additio n t o th e modula r relations (1 .53 ) th e thet a functio n satisfie s th e followin g involutio n equation :

*(£).

If w e assume tha t th e serie s Y^Qi 1 71 )1 71 '1 converge s absolutely , the n thi s formul a is a sligh t modificatio n o f a theorem o f A. Wintner . A s a n example w e appl y (1 .72 ) for f(m) — (/?(n)n -1 whic h i s given b y (1 .36 ) gettin g

(••74 )

£»fi!UJL + + (^ + < >) m^.x oo

= 2x 2 ] T ( 2 + ( - l ) m ) m " 2 + 0 ( x logs) . l

= 3((2)x 2 + 0(x\ogx) =

-(irx) 2 + 0(x\ogx).

This resul t extend s easil y fo r an y fc ^ 4 (writ e r ^ a s th e additiv e convolutio n o f r4 an d r&_4 , appl y th e abov e resul t fo r r^ an d execut e th e summatio n ove r th e remaining fc — 4 squares b y integration ) (1.76) ]

T r fc (n) =

^ 0-+

O f r * " 1 logs) .

Notice tha t thi s improve s th e formul a (1 .71 ) whic h wa s obtaine d b y th e metho d o f packing wit h a uni t square . Th e exponen t | — 1 in (1 .76 ) i s best possibl e becaus e the individua l term s o f summatio n ca n b e a s larg e a s th e erro r ter m (apar t fro m logrr), indeed fo r fc = 4 we have r±(n) ^ 1 6 n if n i s odd b y th e Jacob i formula . Th e only case s o f th e lattic e poin t proble m fo r a bal l whic h ar e no t ye t solve d (i.e. , th e best possibl e erro r term s ar e no t ye t established ) ar e fo r th e circl e (f c = 2 ) an d th e sphere (f c = 3) . W e shal l addres s thes e fascinatin g problem s o n othe r occasions .

23

1. ARITHMETI C FUNCTIO N EXERCISE

3 . Prov e b y th e hyperbol a metho d tha t Y^ r(n 2 + 1) = - z l o gx + 0(x). n^x

1.6. Sum s o f multiplicativ e functions . Throughout / i s a multiplicative function . Sinc e / i s determine d b y it s value s at prim e power s it is possible t o estimat e Mf(x) i n term s o f th e loca l sum s oo

(1-77) *

P (/) = £/&>•>"" •

First w e give simple uppe r bound s whe n / i s non-negative. Th e followin g estimate s need n o explanatio n (1.78) M

f(x)

^xJ2 fWn- 1 ^x]J a

p(f).

One ca n d o bette r i f /i s non-decreasing o n prim e powers . I n thi s cas e h = \x * / i s non-negative, indee d h(p u) = f(jp v) — / ( p ^ - 1 ) ^ 0 if v > 1 . Writin g / = 1 • h w e get

(1.79) M

f(x)

= Y, h(m) [—1 < x J2 fe M™1_ m^x m^x

P^X P^z%

Here i s a crude bu t usefu l applicatio n o f (1.79) fo r f(m) = Tfc(ra)^ . W e hav e f(p) = k l an d

where c = c(k, £) is a positive constant . W e hav e als o th e elementar y boun d

n(i+^)«io g x. p^x

for an y x ^ 2 (se e (2.1 5)) . Henc e (1 .79 ) yield s (1.80) Y^ninY^xilogxf-

1

n^x

where th e implie d constan t depend s o n k,£. Thi s i s a crude bound , bu t o f the correct orde r o f magnitude . Notic e tha t i t implies tha t (1.81) r

k(n)^n

£

for an y e > 0 , th e implie d constan t dependin g o n e and k. We shal l b e usin g thes e bounds fo r th e diviso r functio n r^(n ) ofte n withou t mention .

24

1. ARITHMETI C FUNCTION S

If on e take s th e averag e of f(n) ove r squar e fre e numbers , o r what amount s to the sam e thin g on e assume s / i s supported o n squarefree numbers , the n th e abov e arguments yiel d

(1.82) M

f{x)

^ n l 1 /(P)-I^ P'

p^x

provided f(p) > 1 for all p. W e ma y requir e f(p) ^ 1 to hold tru e fo r primes onl y in a certain set . The n th e resul t is

(i.83) *,