Analyti c Numbe r Theor y

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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

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

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 , 11T23 ,

11T24, 11R42 .

For addi t iona l informatio n an d upda t e s o n th i s book , visi t w w w . a m s . o r g / b o o k p a g e s / c o l l - 5 3

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 0-8218-3633- 1 (acid-fre e paper ) 1. Numbe r theory . I . Kowalski , Emmanuel , 1969 - II . Title . III . Colloquiu m publication s

(American Mathematica l Society ) ; v. 53 .

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Contents

Preface x i

Introduction 1

Chapter 1 . Arithmeti c Function s 9 §1.1. Notatio n an d definition s 9 §1.2. Generatin g serie s 1 0 §1.3. Dirichle t convolutio n 1 2 §1.4. Example s 1 3 §1.5. Arithmeti c function s o n averag e 1 9 §1.6. Sum s o f multiplicative function s 2 3 §1.7. Distributio n o f additiv e function s 2 8

Chapter 2 . Elementar y Theor y o f Prime 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 §3.1. Introductio n 4 3 §3.2. Dirichle t character s 4 4 §3.3. Primitiv e character s 4 5 §3.4. Gaus s sum s 4 7 §3.5. Rea l character s 4 9 §3.6. Th e quarti c residu e symbo l 5 3 §3.7. Th e Jacobi-Dirichle t an d th e Jacobi-Kubot a symbol s 5 5 §3.8. Heck e character s 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 3 §5.1. Definition s an d preliminarie s 9 3

v

vi CONTENT S

§5.2. Approximation s t o L-function s 9 7 §5.3. Countin g zero s o f L-function s 10 1 §5.4. Th e zero-fre e regio n 10 5 §5.5. Explici t formul a 10 8 §5.6. Th e prim e numbe r theore m 11 0 §5.7. Th e Gran d Rieman n Hypothesi s 11 3 §5.8. Simpl e consequence s o f GR H 11 7 §5.9. Th e Rieman n zet a functio n an d Dirichle t L-function s 11 9

§5.10. L-function s o f number fields 12 5 §5.11. Classica l automorphi c L-function s 13 1 §5.12. Genera l automorphi c L-function s 13 6 §5.13. Arti n L-function s 14 1 §5.14. L-function s o f varieties 14 5 §5.A. Appendix : comple x analysi s 14 9

Chapter 6 . Elementar y Siev e Method s 15 3 §6.1. Siev e problems 15 3 §6.2. Exclusion-inclusio n schem e 15 4 §6.3. Estimation s o f V+(z), V~(z) 15 7 §6.4. Fundamenta l Lemm a o f sieve theory 15 8 §6.5. Th e A 2-Sieve 16 0 §6.6. Estimat e fo r th e mai n ter m o f the A 2-sieve 16 4 §6.7. Estimate s fo r th e remainde r ter m i n th e A 2-sieve 16 5 §6.8. Selecte d application s o f A 2-sieve 16 6

Chapter 7 . Bilinea r Form s an d th e Larg e Siev e 16 9 §7.1. Genera l principle s o f estimating doubl e sum s 16 9 §7.2. Bilinea r form s wit h exponential s 17 1 §7.3. Introductio n t o th e larg e siev e 17 4 §7.4. Additiv e larg e siev e inequalitie s 17 5 §7.5. Multiplicativ e larg e siev e inequality 17 9 §7.4. Application s o f the larg e siev e t o sievin g problem s 18 0 §7.6. Panoram a o f the larg e siev e inequalities 18 3 §7.7. Larg e siev e inequalitie s fo r cus p form s 18 6 §7.8. Orthogonalit y o f elliptic curve s 19 2 §7.9. Powe r moment s o f L-functions 19 4

Chapter 8 . Exponentia l Sum s 19 7 §8.1. Introductio n 19 7 §8.2. Weyl' s metho d 19 8 §8.3. Va n de r Corpu t metho d 20 4 §8.4. Discussio n o f exponen t pair s 21 3 §8.5. Vinogradov' s metho d 21 6

Chapter 9 . Th e Dirichle t Polynomial s 22 9 §9.1. Introductio n 22 9 §9.2. Th e integra l mean-valu e estimate s 23 0 §9.3. Th e discret e mean-valu e estimate s 23 2 §9.4. Larg e value s o f Dirichle t polynomial s 23 5 §9.5. Dirichle t polynomial s wit h character s 23 8

TABLE O F CONTENT S vi i

§9.6. Th e reflectio n metho d 24 3 §9.7. Larg e value s o f D(s, x) 24 6

Chapter 10 . Zer o Density Estimate s 24 9 §10.1. Introductio n 24 9 §10.2. Zero-detectin g polynomial s 25 0 §10.3. Breakin g th e zero-densit y conjectur e 25 4 §10.4. Gran d zero-densit y theore m 25 6 §10.5. Th e gap s betwee n prime s 26 4

Chapter 11 . 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 elliptic curve s 29 1

§11.10. Th e loca l zet a functio n o f elliptic curve s 29 7 §11.11. Surve y o f further results : a cohomologica l prime r 30 0 §11.12. Comment s 31 3

Chapter 12 . Characte r Sum s 31 7 §12.1. Introductio n 31 7 §12.2. Completin g method s 31 8 §12.3. Complet e characte r sum s 31 9 §12.4. Shor t characte r sum s 32 4 §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 5

Chapter 13 . Sum s ove r Prime s 33 7 §13.1. Genera l principle s 33 7 §13.2. A variant o f Vinogradov' s metho d 340 §13.3. Linnik' s identit y 34 2 §13.4. Vaughan' s identit y 34 4 §13.5. Exponentia l sum s ove r prime s 34 5 §13.6. Bac k t o th e siev e 34 8

Chapter 14 . Holomorphi c Modula r Form s 35 3 §14.1. Quotient s o f the 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 cusp 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 15 . Spectra l Theor y o f Automorphic 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 length o f closed geodesie s an d clas s number s 40 1

Chapter 16 . Sum s o f Kloosterman Sum s 40 3 §16.1. Introductio n 40 3 §16.2. Fourie r expansio n o f Poincare serie s 40 4 §16.3. Th e projectio n o f Poincar e serie s on Maas s form s 40 6 §16.4. Kuznetsov' s formula s 40 6 §16.5. Estimate s fo r th e Fourie r coefficient s 41 3 §16.6. Estimate s fo r sum s o f Kloosterman sum s 41 5

Chapter 17 . Prime s i n Arithmeti c Progression s 41 9 §17.1. Introductio n 41 9 §17.2. Bilinea r form s i n arithmeti c progression s 42 1 §17.3. Proo f o f the Bombieri-Vinogrado v Theore m 42 3 §17.4. Proo f o f the Barban-Davenport-Halbersta m Theore m 42 4

Chapter 18 . Th e Leas t Prim e i n a n Arithmeti c Progressio n 42 7 §18.1. Introductio n 42 7 §18.2. Th e log-fre e zero-densit y theore m 42 9 §18.3. Th e exceptiona l zer o repulsion 43 4 §18.4. Proo f o f Linnik' s Theore m 43 9

Chapter 19 . Th e Goldbac h Proble m 44 3 §19.1. Introductio n 44 3 §19.2. Incomplet e A-function s 44 5 §19.3. A ternary additiv e proble m wit h A b 44 6 §19.4. Proo f o f Vinogradov' s thre e prime s theore m 44 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 7 §21.1. Weyl' s criterio n 48 7 §21.2. Selecte d equidistributio n result s 48 8 §21.3. Root s o f quadrati c congruence s 49 4 §21.4. Linea r an d bilinea r form s i n quadrati c root s 49 6

TABLE O F CONTENT S i x

§21.5. A Poincare serie s fo r quadrati c root s 49 8 §21.6. Estimatio n o f the 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 §22.3. Th e clas s grou p L-function s 51 1 §22.4. Th e clas s numbe r problem s 51 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 partition 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 lower 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 the Rieman n Ze t a Functio n 54 7 §24.1. A lower boun d fo r N 0(T) 54 7 §24.2. A positive proportio n o f critica l zero s 55 0

Chapter 25 . Th e Spacin g o f the Zero s o f the Rieman n Ze t a-Function 56 3 §25.1. Introductio n 56 3 §25.2. Th e pai r correlatio n o f zero s 56 4 §25.3. Th e n-leve l correlatio n functio n fo r consecutiv e spacin g 57 0 §25.4. Low-lyin g zero s o f L-function s 57 2

Chapter 26 . Centra l Value s o f L-function s 57 7 §26.1. Introductio n 57 7 §26.2. Principl e o f the proo f o f Theore m 26. 2 58 0 §26.3. Formula s fo r th e firs t an d th e secon d momen t 58 2 §26.4. Optimizin g th e mollifie r 58 9 §26.5. Proo f o f Theore m 26. 2 59 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 this book in a period o f time during an d afte r teachin g course s and working with graduate students in Rutgers University, Bordeaux 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 his help 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 boys tried 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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Index

A-process, 204 , 211 , 21 5 ^-process, 204 , 211 , 21 6 L-functions o f varieties , 9 6 A2-sieve, 160 , 43 0 ^-adic cohomology , 26 9 £-adic cohomology groups with compac t sup -

port, 30 4 £-adic sheaf , 31 0 j-invariant, 36 4

Abel-Jacobi Theorem , 299 , 54 3 Abelian variety , 145 , 14 6 Absolute logarithmi c height , 54 1 Absolute values , 54 1 Additive binar y problem , 46 8 Additive character , 44 , 175 , 181 , 271 , 467 ,

475 Additive function , 9 , 2 8 Additive reduction , 36 5 Adjoint square , 137 , 37 3 Algebraic curves , 14 5 Algebraic surface , 31 0 Almost primes , 15 9 Ambiguous classes , 50 9 Ambiguous form , 50 7 Amplification method , 379 , 59 7 Amplifier, 59 6 Analytic L-function , 141 , 143 , 14 9 Analytic conductor , 9 5 Approximate functiona l equation , 97 , 25 7 Arithmetic etal e fundamenta l group , 30 0 Artin L-functions , 96 , 126 , 141 , 356 Artin-Shreier covering , 30 2 Automorphic L-functions , 84 , 93 , 37 5 Automorphic form , 61 , 93, 13 1 Auxiliary polynomial , 28 2 Averaging technique , 357 , 38 7

Bad reduction , 36 4 Bernoulli numbers , 6 7 Bernoulli polynomial , 66 , 48 4 Bessel function, 72 , 73 , 90, 91, 245, 258, 358,

386, 408 , 411 , 454, 51 2 Betti numbers , 305 , 30 7

Bilinear form , 169 , 174 , 186 , 218 , 326 , 340 , 343, 346 , 350 , 42 0

Binary quadrati c forms , 498 , 50 3 Birch an d Swinnerton-Dye r Conjecture , 148 ,

367, 520 , 538 , 578 , 57 9 Brun-Titchmarsh inequality , 167 , 42 0 Buchstab identity , 34 9 Burgess' bound , 53 6

Canonical height , 54 1 Cauchy's inequality , 162 , 169 , 22 0 Central character , 54 0 Central critica l point , 57 7 Central value , 514 , 52 9 Character sum , 74 , 101 , 365, 422 , 42 9 Characters, 43 , 48 7 Chebotarev Densit y Theorem , 143 , 489 Circle method , 171 , 315, 44 3 Class group , 58 , 51 0 Class grou p character , 59 , 131 , 134 , 361 ,

369, 51 6 Class number , 58 , 125 , 368 , 402 , 504 , 523 ,

537, 56 9 Class Numbe r Formula , 38 , 124 , 402 , 513 ,

516 Class numbe r on e problem , 124 , 57 7 Closed point , 29 7 Cohen-Lenstra heuristics , 51 0 Combinatorial sieve , 16 4 Companion sums , 273 , 27 8 Complete L-function , 9 4 Complete exponentia l sum , 45 8 Complete family , 57 3 Complete sums , 269 , 31 8 Completing technique , 31 8 Complex multiplication , 367 , 36 9 Conductor, 45 , 60, 93, 94, 109 , 130 , 149 , 190,

247, 365 , 539 , 572 , 579 , 59 7 Conductor exponent , 36 5 Congruence subgroup , 35 4 Conjugacy classes , 39 4 Constant ter m i n the Fourie r expansion , 39 0 Continuous spectrum , 7 5 Converse theorems , 37 7

612 INDEX

Convexity bound , 101 , 119 , 137 , 244 , 535 , 549, 553 , 59 5

Critical line , 113 , 547 , 56 2 Critical strip , 96 , 101 , 105 , 113 , 14 5 Critical zero , 547 , 56 3 Cubic Gaus s sum , 49 1 Cusp, 355 , 38 7 Cusp form , 61 , 65 , 83 , 136 , 186 , 356 , 368 ,

479, 57 4

Dedekind et a function , 450 , 516 , 54 4 Dedekind multiplie r system , 45 0 Dedekind sum , 450 , 456 , 51 6 Dedekind zet a function , 125 , 51 3 Degree, 9 4 Deligne bound , 291 , 357, 379 , 493 , 53 2 Density conjecture , 232 , 249 , 26 5 Density theorem , 420 , 547 , 59 6 Determinant equation , 38 3 Deuring-Heilbronn phenomenon , 42 8 Difference betwee n consecutiv e primes , 26 6 Differencing process , 201 , 204, 211, 212, 220,

330, 46 6 Diophantine approximation , 28 2 Dirichlet L-functions , 45 , 83 , 84 , 182 , 267 ,

324, 329 , 368 , 388 , 479 , 513 , 534 , 573 , 579

Dirichlet approximatio n theorem , 199 , 45 7 Dirichlet character , 45 , 17 9 Dirichlet convolution , 12 , 59 0 Dirichlet diviso r problem , 21 , 79, 198 , 21 5 Dirichlet polynomial , 194 , 229, 253, 429, 549,

550, 55 4 Dirichlet series , 1 1 Discrete subgroup , 35 4 Discrete valuation , 292 , 30 1 Discriminant, 36 3 Dispersion method , 17 1 Divisor function , 13 , 74 , 86 , 33 4 Divisor o n a curve , 29 3 Dual sum , 170 , 46 8 Duality principle , 170 , 174 , 184 , 189 , 23 4

Effective divisor , 293 , 29 8 Eichler-Shimura theory , 36 7 Eisenstein series , 80 , 357, 360, 369 , 388, 411,

479, 491 , 511, 58 8 Elliptic curve , 145 , 190 , 269 , 313 , 363 , 403 ,

520, 529 , 57 4 Elliptic differentia l operator , 38 6 Elliptic functions , 54 3 Elliptic motion , 39 5 Epstein zet a function , 51 3 Equidistribution, 130 , 137 , 198 , 313 , 352 ,

487 Etale covering , 30 0 Euler characteristic , 30 7 Euler Idonea l Numbe r Problem , 52 0

Euler Pentagona l Number s Theorem , 449 , 456

Euler product , 11 , 41 , 45 , 60 , 94 , 113 , 146 , 297, 366 , 371 , 493, 521 , 529 , 551 , 56 4

Euler-Maclaurin formula , 66 , 68 , 76 , 206 , 483, 48 5

Exceptional character , 37 , 43 4 Exceptional eigenvalues , 390 , 399 , 410 , 49 7 Exceptional zero , 93, 111, 112, 121 , 140, 390,

428, 43 4 Exceptional zer o repulsion , 42 8 Exclusion-inclusion, 145 , 156 , 171 , 308. 33 9 Explicit formula , 108 , 118 , 127 , 265 , 337 ,

410, 440 , 56 4 Exponent pair , 21 4 Exponent Pai r Hypothesis , 86 , 21 4 Exponential integrals , 206 , 20 8 Exponential sums , 197 , 225 , 443 , 45 6

Family o f L-functions , 96 , 175 , 573 , 58 0 Family o f exponentia l sums , 31 2 Farey sequence , 451 , 458, 469 , 47 7 Ford circle , 45 2 Four square s theorem , 46 8 Fourier coefficients , 17 4 Fourier coefficient s o f cus p forms , 319 , 40 4 Fourier coefficient s o f modular forms , 83 , 291,

479 Fourier expansio n a t infinity , 13 2 Fourier integrals , 20 4 Fricke involution , 83 , 366 , 36 8 Frobenius automorphism , 270 , 30 2 Frobenius conjugac y class , 30 2 Function field, 292 , 30 1 Functional equation , 81 , 84 , 94 , 244 , 258 ,

362, 366 , 368 , 375 , 377 , 512 , 530 , 539 , 547, 566 , 58 2

Functional equatio n o f Eisenstein series , 38 9 Functor, 30 1 Fundamental discriminant , 52 , 124, 508, 538,

540, 54 3 Fundamental domain , 58 , 354, 396, 499, 504 ,

521 Fundamental Lemma , 153 , 159 , 43 7 Fundamental unit , 38 , 402 , 51 6

Gamma factor , 9 4 Gauss circl e problem , 20 , 73 , 198 , 21 5 Gauss sum , 18 , 47 , 49 , 60 , 79 , 84 , 119 , 179 ,

192, 199 , 239 , 274 , 321 , 347 , 456 , 474 , 478, 488 , 491 , 495

Gauss-Bonnet formula , 35 5 Gaussian prime , 53 , 130 , 29 0 Gaussian Unitar y Ensemble , 563 , 57 0 Genus character , 362 , 51 6 Genus theory , 480 , 506 , 54 0 Geometric Frobenius , 302 , 30 4 Geometric fundamenta l group , 30 4 Goldbach problem , 171 , 339

INDEX 613

Good reduction , 36 4 Grand Densit y Conjecture , 25 0 Grand Rieman n Hypothesis , 101 , 113 , 136 ,

175, 182 , 194 , 249 , 324 , 419 , 57 7 GRH, 11 3 Gross-Zagier curve , 36 8 Gross-Zagier formula , 57 9 Group la w o n ellipti c curves , 29 1

Haar measure , 48 7 Hadamard an d d e l a Vallee Poussin , 41 , 101,

306 Hankel transform , 71 , 99 Harish-Chandra/Selberg transform , 393 , 397,

501 Harmonic polynomial , 36 2 Harmonics, 9 , 174 , 239 , 319 , 356 , 360 , 38 3 Hasse bound , 193 , 26 9 Hasse derivative , 28 2 Hasse-Weil zet a function , 99 , 145 , 146 , 272 ,

366, 369 , 539 , 541 , 57 9 Hecke L-function , 60 , 86 , 366 , 368 , 374, 379 ,

511, 57 4 Hecke basis , 57 4 Hecke character , 56 , 59 , 129 , 141 , 142 , 302 ,

362 Hecke congruenc e groups , 35 4 Hecke eigenvalues , 134 , 192 , 513 , 58 2 Hecke form , 37 2 Hecke operator , 80 , 186 , 36 7 Heegner point , 494 , 542-544 , 57 9 Heilbronn sums , 30 0 Higher vo n Mangold t function , 16 , 59 2 Hilbert Clas s Field , 54 4 Hilbert inequality , 17 5 Hilbert's Theore m 90 , 29 6 Hybrid larg e sieve , 18 3 Hyperbola method , 22 , 31 , 37, 42 0 Hyperbolic conjugac y classes , 40 1 Hyperbolic geodesies , 38 4 Hyperbolic laplacian , 38 5 Hyperbolic measure , 354 , 51 4 Hyperbolic motion , 39 5 Hyperbolic plane , 38 4 Hyperelliptic curve , 281 , 287

Idoneal number , 50 8 Imaginary quadrati c field , 17 , 56 , 361 , 503 ,

508, 54 2 Incomplete characte r sum , 31 7 Incomplete Eisenstei n series , 387 , 38 9 Incomplete gamm a function , 51 3 Incomplete Kloosterma n sum , 47 1 Invariant integra l operator , 392 , 39 3 Isoperimetric inequality , 39 8

Jacobi inversio n formula , 47 3 Jacobi sum , 49 , 311 , 49 1 Jacobi symbol , 52 , 47 3

Jacobian variety , 147 , 57 8

Kloosterman fractions , 185 , 48 1 Kloosterman sum , 18 , 78, 185 , 187, 278, 308,

313, 322 , 382 , 404 , 469 , 476 , 481 , 492 , 584

Kloosterman sum s zeta-function , 41 2 Kloosterman zet a function , 27 8 Kloosterman-Salie sum , 281 , 358 Kronecker Limi t Formula , 516 , 52 4 Kronecker symbol , 52 , 57 , 124 , 506 , 508 , 530 Kuznetsov formula , 65 , 186 , 38 0

Langlands functoriality , 96 , 369 , 49 3 Langlands program , 13 2 Laplace operator , 89 , 131 , 132 , 186 , 362 ,

404, 563 , 58 0 Large sieve , 164 , 167 , 174 , 194 , 239, 420, 59 5 Large siev e inequality , 125 , 423 , 58 0 Lattice points , 19 7 Least quadrati c non-residue , 18 2 Lefschetz trac e formula , 305 , 31 0 Legendre formula , 155 , 34 1 Legendre symbol , 27 1 Length o f close d geodesies , 41 0 Length spectrum , 40 1 Levinson's method , 55 0 Lindelof Hypothesis , 101 , 116, 186 , 194 , 214,

235, 243 , 256 , 59 7 Linear equivalenc e classes , 29 8 Linear forms , 497 , 49 9 Linear fractiona l transformation , 353 , 50 4 Linearly equivalen t divisors , 29 3 Liouville function , 1 4 Lisse ^-adi c sheaf , 30 1 Local Langland s conjecture , 13 8 Local roo t number , 13 6 Local roots , 9 4 Local zet a function , 36 5 Log-free zero-densit y estimate , 42 8 Low-lying zero , 57 4

Mobius function , 32 , 36 , 42 , 345 , 421 , 430 , 443, 446 , 59 0

Mobius inversion , 13 , 44 , 46 , 15 4 Mobius Randomnes s Law , 338 , 44 4 Maass form , 131 , 38 7 Major arc , 46 7 Matrix coefficients , 48 7 Mellin transform , 61 , 90, 25 7 Mertens formula , 3 4 Minimal model , 36 4 Minor arc , 457 , 462 , 464 , 466 , 46 7 Mixed o f weight s < w, 30 5 Modular curves , 57 7 Modular form , 145 , 35 6 Modular for m o f weigh t one , 35 6 Modular group , 50 3 Modular interpretation , 54 2

614 INDEX

Modular parameterization , 542 , 543 , 54 5 Modularity conjecture , 191 , 366 Mollification, 550 , 562 , 58 1 Mollifier, 25 1 Monodromy group , 313 , 57 3 Mordell-Weil theorem , 57 9 Multiple Kloosterma n sum , 308 , 49 2 Multiplicative characters , 34 , 271 , 308 Multiplicative function , 154 , 165 , 339 , 52 1 Multiplicative inverse , 19 , 5 5 Multiplicative reduction , 36 5 Multiplicity on e principle , 37 3

Near-orthogonality, 170 , 19 2 Nebentypus, 13 1 Negative curvature , 38 4 Newton polyhedron , 308 , 30 9 Norm map , 27 0 Numerus idoneus , 50 8

Order o f vanishin g o f a n L-function , 11 7 Orthogonality o f characters, 35 , 46, 121 , 311 ,

318, 425 , 482 , 49 2 Orthogonality relations , 44 , 45 , 179 , 271 ,

511

Polya-Vinogradov inequality , 325 , 32 6 Pair Correlatio n Conjecture , 266 , 56 9 Parabolic motion , 39 5 Peter-Weyl theorem , 48 7 Petersson formula , 136 , 187 , 188 , 380 , 404 ,

411, 579 , 58 0 Petersson inne r product , 357 , 37 1 Petersson norm , 138 , 542 , 59 5 Poincare duality , 30 5 Poincare metric , 353 , 38 4 Poincare series , 357 , 404 , 50 0 Poincare uppe r half-plane , 353 , 38 4 Point-pair invariant , 39 1 Poisson distribution , 26 6 Poisson summatio n formula , 61 , 69 , 75 , 99 ,

204, 319 , 359 , 391 , 398, 404 , 47 3 Polya-Vinogradov inequality , 523 , 52 4 Positivity, 41 , 105 , 157 , 180 , 37 7 Primary element , 5 4 Prime Numbe r Theorem , 25 , 31 , 110 , 264 ,

401, 419 , 427 , 446 , 448 , 489 , 527 , 56 7 Primes i n arithmeti c progression s t o larg e

moduli, 26 6 Primes i n shor t intervals , 264 , 26 6 Primes splittin g completely , 52 1 Primitive character , 46 , 48 , 70 , 79 , 85 , 119 ,

179, 244 , 422 , 491 , 508, 59 7 Primitive conjugac y classes , 396 , 40 2 Primitive cus p form , 99 , 134 , 370 , 373 , 374 ,

513, 52 9 Primitive Heck e character , 6 0 Primitive ideal , 57 , 50 8 Primitive root , 27 1

Principal divisors , 29 3 Principal genus , 50 6 Pure o f weigh t w, 30 5

Quadratic character , 18 4 Quadratic Reciprocit y Law , 5 1 Quantum uniqu e ergodicit y conjecture , 49 4

Radius o f convergence , 28 9 Ramanujan A function , 360 , 367 , 493 , 51 6 Ramanujan r function , 37 2 Ramanujan sum , 18 , 44 , 48 , 179 , 280 , 323 ,

325, 461 , 472 , 478 , 481 , 483 , 488 , 587 , 589

Ramanujan-Petersson conjecture , 95 , 100, 101, 115, 131 , 137 , 146 , 378 , 391 , 480

Random matri x theory , 56 3 Rankin's trick , 341 , 349, 52 5 Rankin-Selberg L-function , 97 , 118 , 189 , 375,

378, 535 , 58 0 Rankin-Selberg convolution , 14 , 97, 106 , 11 0 Real character , 46 , 4 9 Real primitiv e character , 38 , 46, 57 , 84 , 122 ,

573, 57 7 Real quadrati c field, 38 , 51 6 Reflection method , 243 , 25 7 Regular discriminants , 51 0 Regulator, 12 5 Residual spectrum , 38 8 Resolvent o f th e laplacian , 390 , 40 5 Riemann Hypothesis , 305 , 306, 309, 320, 337,

427, 514 , 518 , 524 , 547 , 563 , 56 8 Riemann Hypothesi s fo r curve s ove r finite

fields, 146 , 329 , 46 3 Riemann Hypothesi s fo r Dirichle t L-functions ,

443 Riemann Hypothesi s fo r ellipti c curve s ove r

finite fields, 36 6 Riemann zet a function , 12 , 119 , 197 , 204 ,

216, 388 , 56 1 Riemann-Roch Theorem , 294 , 296 , 29 8 Root number , 94 , 142 , 145 , 367 , 377 , 538 ,

574 Roots o f a n exponentia l sum , 27 4 Roots o f quadrati c congruences , 5 5

Salie sum , 272 , 323 , 476, 477 , 49 5 Sato-Tate Conjecture , 137 , 289 , 324 , 49 3 Sato-Tate measure , 49 2 Scaling matrix , 355 , 38 7 Scattering matrix , 388 , 39 7 Selberg Eigenvalu e Conjecture , 96 , 390 , 41 5 Selberg sieve , 16 0 Selberg trac e formula , 65 , 39 1 Self-dual L-function , 95 , 106 , 57 8 Semistable, 36 5 Separation o f variables , 326 , 35 0 Series o f Kloosterma n sums , 38 0 Shifted primes , 30 , 15 3

INDEX 615

Short characte r sums , 289 , 32 6 Siegel mas s formula , 48 1 Siegel's bound , 124 , 479 , 51 4 Siegel-Walfisz Theorem , 419 , 427 , 48 9 Sieve dimension , 15 7 Sieve methods , 25 , 153 , 171 , 265 , 340 , 349 ,

430, 49 3 Sieve problem , 2 7 Sieve weights , 33 8 Sieving level , 34 9 Sifted sum , 15 3 Sifting range , 16 1 Sign o f th e functiona l equation , 95 , 53 7 Singular integral , 46 0 Singular series , 460 , 466 , 47 8 Smoothing, 40 , 73 , 74 , 111 , 169 , 239 , 49 7 Special value , 194 , 577 , 59 5 Spectral decomposition , 403 , 406, 50 1 Spectral theory , 19 , 34 4 Spectral theor y o f Kloosterma n sums , 40 3 Square roo t cancellation , 306 , 312 , 31 4 Strict diviso r function , 34 2 Strong multiplicit y on e principle , 139 , 37 5 Subconvexity bound , 101 , 204, 329, 494, 554,

597 Subdivision method , 23 6 Sums o f Kloosterma n sums , 410 , 58 6 Symmetric square , 14 , 137 , 189 , 532 , 535 ,

575, 59 5

Tate twists , 30 1 Tate-Shafarevitch group , 36 7 Tempered diviso r function , 7 5 Theta function , 18 , 61 , 85 , 361 , 456 , 473 ,

479, 513 , 54 1 Theta multiplier , 18 6 Trace map , 27 0 Trivial sheaf , 30 6 Trivial zeros , 9 6 Twisted modula r form , 13 3

Uncertainty principle , 56 4 Unramified, 94 , 11 8 Upper-bound sieve , 43 0

Von Mangold t function , 15 , 31 , 42 Voronoi formula , 39 8

Waring problem , 45 6 Weierstrass equation , 363 , 54 3 Weil bound , 79 , 130 , 269, 288, 381, 403, 413,

472, 58 5 Weil conjectures , 37 8 Well-factorable function , 42 0 Well-spaced, 175 , 218 , 23 2 Weyl Law , 186 , 39 1 Weyl shift , 216 , 31 4 Weyl sum , 198 , 201 , 335

Zero-density theorem , 249 , 26 4 Zero-detecting polynomials , 25 2 Zero-free region , 105 , 110 , 128 , 135 , 138 ,

249, 264 , 336 , 347 , 428 , 42 9 Zeta functio n o f a n exponentia l sum , 27 3

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Titles i n Thi s Serie s

53 Henry k Iwanie c an d Emmanue l Kowalski , Analyti c numbe r theory , 200 4

52 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d symplecti c topology ,

2004

51 Alexande r Beil inso n an d Vladimi r Drinfeld , Chira l algebras , 200 4

50 E . B . Dynkin , Diffusions , superdiffusion s an d partia l differentia l equations , 200 2

49 Vladimi r V . Chepyzho v an d Mar k I . Vishik , Attractor s fo r equation s o f mathematical physics , 200 2

48 Yoa v Benyamin i an d Jora m Lindenstrauss , Geometri c nonlinea r functiona l analysis ,

Volume 1 , 200 0

47 Yur i I . Manin , Frobeniu s manifolds , quantu m cohomology , an d modul i spaces , 199 9

46 J . Bourgain , Globa l solution s o f nonlinea r Schrodinge r equations , 199 9

45 Nichola s M . Kat z an d Pete r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 199 9

44 Max-Alber t Knus , Alexande r Merkurjev , an d Marku s Rost , Th e boo k o f

involutions, 199 8

43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 199 5

42 Victo r Guil lemi n an d Shlom o Sternberg , Variation s o n a them e b y Kepler , 199 0

41 Alfre d Tarsk i an d Steve n Givant , A formalizatio n o f se t theor y withou t variables , 198 7

40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3

39 N . Jacobson , Structur e an d representation s o f Jorda n algebras , 196 8

38 O . O r e , Theor y o f graphs , 196 2

37 N . Jacobson , Structur e o f rings , 195 6

36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5

35 A . C . SchaefFe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0

34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 195 0

33 J . F . Ri t t , Differentia l algebra , 195 0

32 R . L . Wilder , Topolog y o f manifolds , 194 9

31 E . Hill e an d R . S . Phill ips , Functiona l analysi s an d semigroups , 195 7

30 T . Rado , Lengt h an d area , 194 8

29 A . Weil , Foundation s o f algebrai c geometry , 194 6

28 G . T . Whyburn , Analyti c topology , 194 2

27 S . Lefschetz , Algebrai c topology , 194 2

26 N . Levinson , Ga p an d densit y theorems , 194 0

25 Garret t BirkhofF , Lattic e theory , 194 0

24 A . A . Albert , Structur e o f algebras , 193 9

23 G . Szego , Orthogona l polynomials , 193 9

22 C . N . Moore , Summabl e serie s an d convergenc e factors , 193 8

21 J . M . Thomas , Differentia l systems , 193 7

20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n th e comple x

domain, 193 5

19 R . E . A . C . Pale y an d N . Wiener , Fourie r transform s i n th e comple x domain , 193 4

18 M . Morse , Th e calculu s o f variation s i n th e large , 193 4

17 J . M . Wedderburn , Lecture s o n matrices , 193 4

16 G . A . Bliss , Algebrai c functions , 193 3

15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932

14 J . F . Ri t t , Differentia l equation s fro m th e algebrai c standpoint , 193 2

TITLES I N THI S SERIE S

13 R . L . Moore , Foundation s o f poin t se t theory . 193 2

12 S . Lefschetz , Topology , 193 0

11 D . Jackson , Th e theor y o f approximation , 193 0

10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9

9 G . D . Birkhoff , Dynamica l systems , 192 7

8 L . P . Eisenhart , Non-Riemannia n geometry , 192 7

7 E . T . Bell , Algebrai c arithmetic , 192 7

6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927

5.1 G . C . Evans , Functional s an d thei r applications ; selecte d topics , includin g integra l equations, 191 8

5.2 O . Veblen , Analysi s situs , 192 2

4 L . E . Dickson , O n invariant s an d th e theor y o f number s

W . F . Osgood , Topic s i n th e theor y o f function s o f severa l comple x variables , 191 4

3.1 G . A . Bliss , Fundamenta l existenc e theorems , 191 3

3.2 E . Kasner , Differential-geometri c aspect s o f dynamics , 191 3

2 E . H . Moore , Introductio n t o a for m o f genera l analysi s

M. Mason , Selecte d topic s i n th e theor y o f boundar y valu e problem s o f differentia l equations

E. J . Wilczynski , Projectiv e differentia l geometry , 191 0

1 H . S . Whi te , Linea r system s o f curve s o n algebrai c surface s

F . S . Woods , Form s o n noneuclidea n spac e

E. B . Va n Vleck , Selecte d topic s i n th e theor y o f divergen t serie s an d o f continue d fractions, 190 5