Other Title s i n Thi s Serie s

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y

and harmoni c analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y

value problems , 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d

the stud y o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Wil l ia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m

mechanics, 198 7

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n ,

1986

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Al le n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups ,

1985

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s

to invarian t subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Will ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4 (Continued in the back of this publication)

This page intentionally left blank

Ten Lectures on the Interfac e

Between Analytic Number Theory and Harmonic Analysis

This page intentionally left blank

Conference Boar d of the Mathematical Science s

CBMS Regional Conference Serie s in Mathematic s

Number 8 4

Ten Lectures on the Interfac e

Between Analytic Number Theory

and Harmonic Analysis

Hugh L . Montgomer y

Published fo r th e Conference Boar d o f the Mathematica l Science s

by th e American Mathematica l Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/084

Exposi tory Lecture s

from th e NSF-CBM S Regiona l Conferenc e

held a t Kansa s S ta t e University , M a n h a t t a n , Kansa s

May 22-25 , 199 0

Research part ial l y suppor te d b y

National Scienc e Foundat io n Gran t DM S 891291 7

1991 Mathematics Subject Classification. Primar y 11-02 , 42-02 ; Secondary 11K06 , 11K38 , 11K70 , 11L03 , 11L07 , 11L15 , 11M20 , 11M26 , 11N05 , 11N25 ,

11N30, 11N69 , 11R06 , 41A30 , 42A05 , 42A10 .

L i b r a r y o f Congres s Ca ta log ing - in -Pub l i ca t i o n D a t a

Montgomery, Hug h L . (Hug h Lowell) , 1944 -Ten lecture s o n th e interfac e betwee n analyti c numbe r theor y an d harmoni c analysis/Hug h L .

Montgomery. p. cm . — (Regiona l conferenc e serie s i n mathematics , ISS N 0160-7642 ; no . 84 )

"Expository lecture s fro m th e NSF-CBM S regiona l conferenc e hel d a t Kansa s Stat e University , Manhattan, Ma y 22-25 , 1990"—T.p . verso .

Includes bibliographica l references . ISBN 0-8218-0737- 4 1. Numbe r theory—Congresses . 2 . Harmoni c analysis—Congresses . I . Title . II . Series .

QA1.R33 no . 8 4 [QA241] 510 s—dc20 94-2686 4 [5i2'73] C I P

C o p y i n g a n d r e p r i n t i n g . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for 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 any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionQams.org .

© Copyrigh t 199 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 .

t<9 Printed o n recycle d paper . This volum e wa s printe d directl y fro m cop y prepare d b y th e autho r usin g A^yfS-T^i,

the America n Mathematica l Society' s T j X macr o system .

10 9 8 7 6 5 4 3 2 9 9 9 8 9 7 9 6

Contents

Preface x i

Notation xii i

Chapter 1 . Unifor m Distributio n 1 1. Qualitativ e theor y 1 2. Quantitativ e relation s 3 3. Trigonometri c approximatio n 1 0 4. Note s 1 3

References 1 5

Chapter 2 . va n de r Corpu t Set s 1 7 Introduction 1 7 Extremal measure s 2 3 Relations betwee n a , /?oo > #2 2 5 Corollaries 2 8 A sufficien t conditio n 3 1 Intersective set s 3 4 Heilbronn set s 3 5 Notes 3 7 References 3 7

Chapter 3 . Exponentia l Sum s I : The Method s o f Wey l an d va n de r Corpu t 3 9 1. Introductio n 3 9 2. WeyP s metho d 3 9 3. va n de r Corput' s metho d 4 6 4. Exponen t pair s 5 6 5. Note s 6 0

References 6 1

Chapter 4 . Exponentia l Sum s II : Vinogradov's Metho d 6 5 1. Introductio n 6 5 2. Vinogradov' s Mea n Valu e Theore m 6 9 3. A bound fo r Wey l sum s 7 6

CONTENTS

4. A n alternativ e derivatio n 7 9 5. Note s 8 1

References 8 2

Chapter 5 . A n Introductio n t o Turan' s Metho d 8 5 1. Introductio n 8 5 2. Turan' s Firs t Mai n Theore m 8 6 3. Fabry' s Ga p Theore m 8 9 4. Longe r range s o f v 9 1 5. Turan' s Secon d Mai n Theore m 9 3 6. Specia l coefficient s b n 9 7 7. Note s 10 2

References 10 5

Chapter 6 . Irregularitie s o f Distributio n 10 9 1. Introductio n 10 9 2. Square s 11 0 3. Disk s 11 1 4. Deca y o f the Fourie r Transfor m 11 4 5. Familie s allowin g translation , scalin g an d rotatio n 11 9 6. Note s 12 0

References 12 2

Chapter 7 . Mea n an d Larg e Value s of Dirichle t Polynomial s 12 5 1. Introductio n 12 5 2. Mea n value s vi a trigonometri c approximatio n 12 7 3. Majo r ant principle s 13 1 4. Revie w o f Elementar y Operato r Theor y 13 4 5. Mea n value s vi a Hilbert' s inequalit y 13 7 6. Larg e valu e estimate s 14 0 7. Note s 14 3

References 14 6

Chapter 8 . Distributio n o f Reduce d Residue Classe s i n Shor t Interval s 15 1 1. Introductio n 15 1 2. A probabilisti c mode l 15 3 3. A n approac h b y Fourie r technique s 15 4 4. Th e fundamenta l lemm a 15 6 5. Note s 16 0

References 16 1

Chapter 9 . Zero s o f L- functions 16 3 1. Introductio n 16 3 2. Leas t Characte r Non-Residue s 16 4

CONTENTS IX

3. Clump s o f zeros 16 8 4. Th e Deuring-Heilbron n Phenomeno n 17 2 5. Note s 17 6

References 17 7

Chapter 10 . Smal l Polynomial s with Integra l Coefficient s 17 9 1. Introductio n 17 9 2. Th e Gorskov-Wirsin g Polynomial s 18 3 3. Note s 18 8

References 19 0

Appendix: Som e Unsolve d Problem s 19 5 1. Unifor m Distributio n 19 5 2. va n de r Corpu t Set s 19 6 3. Wey l Sum s 19 6 4. va n de r Corput' s Metho d 19 7 5. Turan' s Metho d 19 7 6. Irregularitie s o f Distribution 19 8 7. Mea n an d Larg e Value s o f Dirichle t Polynomial s 19 8 8. Reduce d Residue s i n Shor t Interval s 20 0 9. Zero s o f L-Function s 20 1 10. Smal l Polynomial s wit h Integra l Coefficient s 20 1 11. Characte r Sum s 20 2 12. Diophantin e Approximatio n 20 2 13. Metri c Diophantin e Approximatio n 20 4 14. Algebrai c Integer s 20 5 15. Trigonometri c Polynomial s 20 6 16. Miscellaneou s 20 7

References 21 0

Index 215

This page intentionally left blank

Preface

Todd Cochran e an d Rober t Dressie r note d th e commo n interest s o f numbe r theorists an d harmoni c analyst s i n thei r department , an d propose d a n NSF -CBMS conferenc e whos e purpos e woul d b e t o furthe r th e cross-fertilizatio n be -tween thes e tw o areas . Th e conferenc e too k plac e durin g th e wee k Ma y 21-25 , 1990, o n th e campu s o f Kansa s Stat e University , i n Manhatta n ("Th e Littl e Apple"), Kansas . Th e conferenc e wa s funde d b y NS F Gran t DMS-8912917 . In addition , NS A Gran t MDS-904-90-14-401 8 provide d th e trave l expense s o f a number o f graduate s students , enablin g the m t o atten d th e conference .

In additio n t o th e lecture s reporte d upo n i n thi s volume , th e conferenc e fea -tured th e followin g nin e majo r addresses :

Speaker

J. Bec k (Rutgers U. )

P. X . Gallaghe r (Columbia U. )

H. Iwanie c (Rutgers U. )

T. W . Korne r (Cambridge U. )

J.-F. Mel a (U. Pari s XIII )

I. Z . Ruzs a (MTA, Budapest )

B. Saffar i (U. Pari s XI )

G. Tenenbau m (U. Nancy )

R. C . Vaugha n (Imperial College )

Title

New Applications of the Roth-Haldsz Method in Irregularities of Distribution

Duality for long second Moments of Short Interval Densities

Estimates for coefficients of L-functions

Universal Fourier Series

Remarkable Subgroups of the Circle

Nouveaux sets of Trigonometric Polynomials: Halfway between Random and Regular

An Account of extremal Problems on Polynomials with restricted Coefficients

Fourier Transforms of Divisor Distributions: A Survey of Applications

The Upper Bound for G(k) in Waring's Problem

xii PREFAC E

Even durin g th e conference , th e cross-fertilizatio n pai d dividends : Beck , afte r hearing Saffari' s talk , realize d tha t h e had tool s at hi s disposal tha t enable d hi m to contribut e t o a n ol d proble m concernin g trigonometri c polynomial s (se e th e commentary o n Proble m 55 . o n p . 206) .

The conference wa s enhanced b y several socia l events , including a n invigorat -ing excursio n t o th e Konz a Prairie . O n behal f o f th e 8 8 participants , i t i s m y pleasure t o than k th e organizer s fo r thei r fin e work , an d th e mathematician s a t KSU fo r thei r hospitality .

Most of the material in this volume is found i n the research literature, and does not originat e fro m th e presen t author . Origina l result s publishe d her e fo r th e first tim e were obtained wit h the support fro m th e Nationa l Scienc e Foundation , particularly NS F Gran t DMS-9107605 .

The autho r sough t an d receive d assistanc e concernin g th e content s o f thi s volume from man y associates . Fo r their suggestions , larg e or small , the author i s indebted t o them all , most especiall y to R . C . Baker , P . T. Bateman , J . Beck , J . Bourgain, P . Bundschuh, T . Cochrane , J . B . Conrey, R . Cook , R . E . Dressier , P . Enflo, J . B . Friedlander, S . W. Graham , G . Halasz, P . R. Halmos , G. Harman, H . Iwaniec, T . W. Korner , R . Lyons , M. Mendes-France , R . Nair , A . D . Pollington , J. B . Rauch , I . Z . Ruzsa , B . SafTari , J . W . Sander , A . Schinzel , J.-P . Serre , G . Tenenbaum, R . C . Vaughan , E . Wirsing , an d T . D . Wooley .

Hugh L . Montgomer y Ann Arbo r 13 May, 199 4

Notation

The harmoni c analysi s pursue d her e involve s th e rea l lin e K, th e circl e grou p T = R/Z , o r the finite Fourie r transform (calle d "additiv e characters" b y number theorists). W e le t

e(x) = e 2nlx

denote th e comple x exponentia l wit h perio d 1 , s o tha t i f / £ L 1(T) the n it s Fourier coefficient s ar e give n b y th e formul a

/(*;)= [ f(x)e(-kx)dx.

Similarly, th e Fourie r coefficient s o f a Bore l measur e / i on T w e take t o b e

fi(k) = / e(—kx)d/j,(x).

We le t [x] and {x} denot e th e integra l par t o f x an d th e fractiona l par t o f x, respectively. Thu s x = [x] + {x} wit h [x] € Z an d 0 < {x} < 1 . I n addition , w e let \\x\\ denote th e distanc e fro m x t o th e neares t integer , ||x| | = min nGz \x — n\. Thus ||x| | i s the natura l nor m o n T .

The relatio n / < C g mean s exactl y th e sam e thin g a s / = 0(g); tha t is , there i s a n absolut e constan t C suc h tha t | / | < Cg fo r al l value s o f th e fre e variables unde r consideration . I f th e implici t constan t C i s allowe d t o depen d on a parameter k, the n suc h dependence ma y b e indicated b y writing / <Cf c g or / = O k(g).

More specialize d notation , appropriat e t o variou s topics , i s develope d i n in -dividual chapters . Suc h notatio n shoul d no t b e expecte d t o b e consisten t fro m one chapte r t o another . Fo r example , 6 in Chapte r 1 is quite differen t fro m 6 in Chapter 2 .

Xlll

This page intentionally left blank

This page intentionally left blank

Index

A boldface numbe r indicate s a princip a

A

Aardenne-Ehrenfest, T . va n 120 , 122 Abramowitz, M . 114 , 12 2 adjoint 13 4 almost periodi c 12 5 Alon, N . 3 7 Andria, G . D . 19 0 Ankeny, N . C . 176 , 17 7 Aparicio, E . 188-19 1 Arkhipov, G . I . 8 2 asymptotic densit y

upper 3 4 Atkinson, F . V . 104 , 10 5

B

Bachelis, G . F . 144 , 14 6 Baker, R . C . vi , 37 , 81, 82, 160 , 204 Balkema, A . A . 103-10 5 Barker polynomia l 20 7 Barnsley, M . F . 189 , 19 1 Bateman, P . T . v i Beck, J . v , vi , 120 , 122 , 123 , 195 , 198,

206, 21 0 Bernstein's inequalit y 2 6 Berry, M . V . 6 1 Bertin, M . J . 205 , 210 Bertrandias, J.-P . 37 , 38 Bertrand-Mathis, A . 37 , 38 Bessel functio n 111-11 4 Beurling, A . 1 4

function 1 4 polynomial 5 , 1 4

binary weigh t 20 8 Biro, A . 10 4 Blanksby, P . E . 104 , 10 5

2

reference, o r a reference t o a definition .

Boas, R . P . 144 , 14 6 Bohr, H . 20 6 Bombieri, E . 59 , 61, 62, 81 , 82, 176 ,

177 Bourgain, J . vi , 34 , 37 , 38 , 142 , 14 6 Boyd, D . W. 205 , 210 Bruijn, N . G . d e 103 , 105 Bundschuh, P . vi , 18 8 Byrnes, J . S . 206 , 210

C

Cantor, D . C . 205 , 210 Cassels, J . W . S . 14 , 15 , 104 , 105 ,

202-204, 21 0 Chandrasekharan, K . 14 , 15 , 81, 82 character sum s 20 2 Chebyshev, P . L . 188 , 19 1

inequality fo r polynomial s 9 5 prime numbe r estimate s 45 , 179 ,

188 Chen, W . W . L . 120-123 , 198 , 210 Chudakov, N . G . 176 , 17 7 Chung, Fa n 20 2 circle grou p 1 Cochrane, T . v , v i Cohn, H . 20 2 Conrey, J . B . v i Cook, R . v i Cordoba, A . 20 7 Corput, J . G . va n de r 17 , 18 , 20, 38,

60, 61 , 120, 12 3 Lemma 18-20 , 39 , 46

Generalized 1 8 method 39 , 46-61 , 19 7

216 INDEX

van de r Corpu t Process A 46 , 5 7 Process B 46, 47, 54-5 7 set 17-38 , 19 6 Theorem 1 8

Generalized 2 0 Coutsias, E . A . 61 , 62 covering congruence s 20 9 Cramer, H . 151 , 161

conjecture 151 , 16 0 critical lin e 16 4 critical stri p 16 5 Cusick, T . W . 203 , 204, 210 , 21 1

D

Danzer, L . 19 8 Davenport, A . 15 , 83 Davenport, H . 33 , 38, 101 , 105, 130,

146, 166 , 168 , 177 , 203 , 211 Davies, R . O . 14 6 Decomps-Guilloux, A . 205 , 210 Dekking, F . M . 61 , 62 Delange, H . 3 7 Deshouillers, J.-M . 61 , 62 Deuring, M . 164 , 176 , 17 7

-Heilbronn phenomeno n 164 , 172176

Diamond, H . 188 , 191 , 204, 21 1 difference se t 3 4 Diophantine approximatio n 202-20 4 Dirac measur e 2 2 Dirichlet, P . G . Lejeun e

character 101 , 146, 164-16 7 divisor proble m 56 , 5 9 L- function

non-trivial zero s o f 16 5 trivial zero s o f 16 5 zeros o f 163-178 , 20 1

polynomial 125-14 9 generalized 126 , 20 6 large value s o f 140-143 ,

146, 198-20 0 mean value s o f 127-131 , 140 ,

143-146, 198-20 0 theorem o n Diophantin e approximation 35 , 85 , 99, 20 3

discrepancy 2-10 , 14 , 109-114 , 119-122, 195 , 198

distribution irregularities o f 109-124 , 164 , 195,

198 for disk s 110 , 111-11 4 for square s 110-11 1

uniform 1-15 , 17-22 , 33 , 34, 37, 125, 19 5

Dobrowolski, E . 205 , 211 Dressier, R . E . v , v i Drmota, M . 19 1 duality 20 , 103 , 104 , 134 , 14 1 Duffin, R . J . 204 , 21 1 Dumir, V . C . 203 , 211

E

Elkies, N . 10 4 Ellison, W . J . 81 , 82 Elliott, P . D . T . A . 202 , 211 Enflo, P . vi , 14 4 equichordal 20 9 Erdos, L . 10 6 Erdos, P . 14 , 15 , 105 , 106 , 121 , 123,

144, 146 , 160 , 161 , 188, 191, 199-201, 203 , 204, 206 , 207, 209-211

-Turan inequalit y 8-10 , 14 , 27 , 19 5 Euler, L .

identity 95 , 96 product formul a 16 8

exponent pair s 39 , 56-61 , 19 7

F

Fabry, E . 9 0 gap theore m 86 , 89-91 , 10 3

Fabrykowski, J . 10 5 Falconer, K . J . 14 6 FejeYs kerne l 5 , 18 , 35, 92, 98, 101,

105, 17 3 Fejer's theore m 19 , 20 , 25 Ferguson, L e Baron O . 19 1 Fourier transfor m 1

rate o f decay 110 , 114-11 9 Fresnel's integra l 49 , 5 5 Friedlander, J . B . vi , 202 , 211

INDEX 217

Fuchs, W . H . J . 144 , 14 6 Fujii, A . 14 6 Fiirstenberg, H . 19 6

G

Gaier, D . 103 , 106 Gallagher, P . X . v , 145-147 , 176 , 177 ,

204, 21 1 Gauss su m 10 , 10 1 Gelfond, A . O . 188 , 191 , 192, 208 Geronimo, J . S . 189 , 19 1 Gilbreath, N . L . 20 8 Goldberg, J . 6 1 Gonek, S . M . 104 , 10 6 Gorskov, D . S . 187 , 189 , 19 2

polynomial 183-19 0 Graham, R . L . 21 0 Graham, S . W. vi , 60-62 , 176 , 17 7 Grandet-Hugot, M . 205 , 210 Green's function s 10 4 Green's theore m 114 , 11 6 Gronwall, T . H . 176 , 17 7

H

Helly selection 3 2 Hensley, D . 204 , 21 1 Hermitian 136 , 137 , 13 9 Herz, C . S . 122 , 123 Herzog, F . 20 8 Hewitt, E . 19 2 Hilbert, D . 137 , 145

inequality 137-140 , 143 , 145 , 19 9 Hildebrand, A . 16 1 Hindry, M . 104 , 106 Hochwald, S . 145 , 147 Hooley, C . 160 , 16 1 Hunt's theore m 8 2 Hua, L . K . 81 , 82 Huxley, M . N . 60-62 , 146-14 8

I

intersective se t 23 , 34, 35 , 37 Ingham, A . E . 8 1 Iwaniec, H . v , vi , 59 , 61, 63, 200, 202 ,

211

J

Jarnik, V . 203 , 212 Julia se t 18 9 Jutila, M . 144 , 146 , 148 , 176 , 17 7

K

Kahane, J.-P . 206 , 21 2 Kamae, T . 37 , 38 Karatsuba, A . A . 14 , 15 , 81, 82 Katznelson, Y . 209 , 212 Kaufman, R . 204 , 212 Kazarinoff, N . D . 61 , 62 Kendall, D . G . 122 , 12 3 Khintchin, A . 204 , 21 2 Knapowski, S . 103 , 106 , 176 , 17 7 Koksma, J . F . 14 , 15 , 60, 6 1

inequality 14 , 2 8 Kolesnik, G . 60-62 , 104 , 10 6 Komlos, J . 19 8 Korneichuk, N . 95 , 10 6 Korner, T . W . v , vi , 206 , 208 , 212 Korobov, N . M . 82 , 83 , 176 , 17 7 Khz, I . 37 , 38 Kronecker's theorem 37 , 85 , 12 5 Kuipers, L . 14 , 1 5

Hadamard ga p 2 3 Hajela, D . J . 20 7 Halasz, G . vi , 104 , 106 , 121-123 , 144,

146, 147 , 197 , 200, 21 1 Halmos, P . R . vi , 14 4 Halton, J . H . 121 , 123 Hankel, H . 113 , 123 Hans-Gill, R . 203 , 211 Hardy, G . H . 4 , 119 , 123 , 133 , 144,

145, 147 , 207 -Littlewood Tauberia n theore m 11 9

Harman, G . vi , 160 , 204 Harrington, A . N . 189 , 19 1 Hausdorff dimensio n 146 , 196 , 203,

204 Hausman, M . 160 , 16 1 Heath-Brown 60-63 , 146 , 147 , 176,

177 Heilbronn, H . 35 , 38, 164 , 176 , 177,

211 set 23 , 35-37, 19 6 theorem 35 , 37 , 19 6

218 INDEX

L

Lachance, M . M . 104 , 106 , 192 Lagarias, J . C . 176 , 17 7 Landau, E . 4 , 103 , 106 , 208 Lang, S . 104 , 10 6

height conjectur e 10 4 lattice constan t 20 3 least characte r non-residu e 164-167 ,

176 Lehmer, D . H . 205 , 212 LeVeque, W . J . 9 , 15 , 203, 211

inequality 9 , 10 , 1 4 Levinson, N . 176 , 17 7 Lindelof Hypothesi s 59 , 141 , 142 linear programmin g 20 , 21 Linnik, Ju . V . 81-83 , 172 , 176 , 17 7

constant 16 3 lemma 71-72 , 74 , 82

Littlewood, J . E . 102 , 103 , 104 , 106 , 133, 144 , 145 , 147 , 148 , 176, 178, 202 , 203, 206-208

Logan, B . F . 144 , 14 8 Lorch, L . 104 , 10 6 Lou, S . T . 160 , 16 2 Louboutin, R . 205 , 212 Loxton, J . H . 60 , 6 3 Lukasenko, S . Ju . 208 , 212 Luquin, F . 19 2 Lyons, R . v i

M

Mack, J . 18 8 Mahler measur e 20 5 Maier, H . 160-16 2 majorant principle s 131-134 , 143 ,

144 Makai, E . 103 , 106 Markov inequalit y 181 , 188 Mela, J.-F . v Mendes-France, M . vi , 37 , 38 , 61-63 ,

81, 8 2 Michelacci, G . 209 , 212 Min, S . H . 6 1 Minkowski's conve x bod y theore m

105 moments 76 , 125 , 151

Montgomery, H . L . 14 , 37 , 82 , 104 -106, 120 , 122 , 123 , 143-148, 148, 160 , 162 , 176-178 , 188 , 197, 199 , 202, 211 , 212

Moore, R . R . 61 , 63 Mordell, L . J . 65 , 68, 71 , 77, 82, 83 Moser, L . 20 4 Motohashi, Y . 176 , 17 8 Mozzochi, C . J . 61 , 63

N

Nair, M . 188 , 19 2 Nair, R . v i Newman, D . J . 206 , 20 7 Newton-Girard identitie s 66^ 71 , 72,

97 Niederreiter, H . 14 , 15 Nieland, L . W . 60 , 63 normal matri x 13 6 normal numbe r 202-20 4 numerical radiu s 135-13 7

O

Odlyzko, A . M . 176 , 177 , 208 , 212 Olivier, M . 208 , 212 operator

norm 134-13 7 theory 134-14 0

Ostrowski, A . 120 , 12 3

P

Paley, R . E . A . C . 103 , 107 Parseval's identit y 4 , 109 , 13 3

approximate 115 , 126 , 12 8 Pathiaux-Delefosse, M 205 , 210 Peres, Y . 37 , 38 Perron's formul a 13 0 Phillips, E . 61 , 63 Pintz, J . 176 , 17 8 Piranian, G . 20 8 Plancherel's theore m 11 6 Poincare recurrenc e 3 7 Poisson summatio n formul a 14 , 39,

46, 4 9 truncated 49 , 60 , 61 , 141

Pollington, A . D . vi , 204 , 212

INDEX 219

Polya, G . 145 , 14 7 -Vinogradov inequalit y 10 , 14 6

Pomerance, C . 160 , 162 , 203 , 211 Poorten, A . J . va n de r 61 , 63 Postnikov, A . G . 176 , 17 8 Powell, M . J . D . 95 , 107 power su m 85-107 , 164 , 17 2 Preissmann, E . 145 , 148 prime number theore m 18 0 primes 33 , 151

Gaussian 20 7 Proth, F . 20 8 PV numbe r 20 5

R

Rahman, Q . I . 104 , 10 7 Ramachandra, K . 146 , 14 9 random variable s 125 , 153 , 209 Rankin, R . A . 61 , 63, 160 , 16 2 Rauch, J . B . v i Rausch, U . 205 , 212 rearrangement o f Fourier coefficient s

144 reduced residu e classe s 151-162 , 200 ,

201 Renyi, A . 105 , 10 6 resultant 18 1 Riemann, G . R . B .

Hypothesis 163 , 20 1 Generalized 163 , 164 , 167 , 20 1

zeta functio n 59 , 60, 82 , 141 , 161, 168-172, 17 6

Riesz produc t 2 3 Riesz representation theore m 2 4 Ringrose, C . 176 , 17 7 Rivlin, T . J . 95 , 10 7 Rodosskii, K . A . 176 , 17 8 Roth, K . F . 15 , 83, 120-12 4 Rozin, S . M . 176 , 17 8 Russell, D . 104 , 10 6 Ruzsa, I . Z . v, vi , 14 , 15 , 37, 38, 104 ,

122, 124 , 195 , 198 , 199 , 212, 213

S

Saff, E . B . 104 , 106 , 19 2 Saffari, B . v , vi , 20 7 Salem, R . 205 , 213

numbers 20 5 Sander, J . W . v i Sanov, I . N . 19 2 Schaefer, H . H . 24 , 38 Schaeffer, A . C . 188 , 192 , 204, 21 1 Schafke, R . 209 , 213 Schinzel, A . v i Schmeisser, G . 104 , 10 7 Schmidt, W . M . 37 , 38 , 120 , 121 , 124,

202, 203 , 211, 213 Schreiber, J . P . 205 , 210 Schur, I . 137 , 145 , 149 Selberg, A . 14 , 15 , 145

functions 14 , 127-129 , 14 3 polynomials 6 , 14 , 12 7

Selfridge, J . L . 20 9 Serre, J.-P . vi , 17 6 Shapiro, H . N . 160 , 16 1 Shapiro, H . S . 144 , 149 , 208 Shnirelman, L . G . 18 8 Sidon se t 10 5 Siegel, C . L . 105 , 107 , 205

theorem 17 6 Silverman, J . 104 , 10 6 skew-Hermitian 13 9 Skubenko, B . F . 202 , 21 3 Smyth, C . J . 105 , 107 , 205 , 213 spectral radiu s 13 5 Sprindzhuk, V . G . 204 , 21 3 squarefree number s 5 6 squares 21 , 31, 33, 196 , 20 7

sums o f two 20 8 stationary phas e 47 , 51-53 , 61 , 115 Stechkin, S . B . 82 , 8 3 Stegun, I . 114 , 12 2 Stenger, F . 104 , 10 7 Stein, P . 10 4 Straus, E . G . 104 , 106 , 205 , 210 Suranyi, J . 10 7 Swinnertion-Dyer, H . P . F . 203 , 210 Szalay, M . 10 7 Szego, G . 95 , 10 7

220 INDEX

Szekeres, G . 20 9

T Tenenbaum, G . v , vi , 200 , 213 Tijdeman, R . 103-105 , 107 , 121 , 124 Titchmarsh, E . C . 60 , 61 , 63, 81, 83,

131, 143 , 149, 176 , 17 8 Trigub, R . M . 192 , 19 3 Turan, P . 14 , 15 , 102-104, 107 , 146,

147 First Mai n Theore m 86-89 , 91 , 96,

103, 19 7 Second Mai n Theore m 86 , 93-9 7 method 85-107 , 176 , 19 7

Tyrina, O . V . 82 , 8 3

U

Uchiyama, S . 104 , 10 7 Ullman, J . L . 188 , 19 2 unitary similarit y 13 6 unitary 136 , 13 7

V

Vaaler, J . D . 14 , 15 , 143 , 145 , 148, 149, 195 , 204, 211 , 213

function 1 5 Lemma 6 , 10-1 4 polynomial 5 , 1 4

Vallee Poussin , Ch . d e l a 17 6 Varga, R . S . 104 , 106 , 19 2 Vaughan, R . C . v , vi , 60 , 63 , 81, 83,

143, 145 , 148 , 160 , 162 , 199, 202, 204 , 21 2

Veech, W . A . 20 9 VilcinskiT, V . T . 204 , 21 3 Vinogradov, I . M . 4 , 14 , 15 , 33, 46,

65, 81 , 83, 176 , 17 8 hypothesis 163 , 164 , 17 6 little glasse s 1 4 mean valu e theore m 67-78 , 19 7 method 46 , 65-83 , 17 6

Volcic, A . 209 , 21 2 Volkmer, H . 209 , 21 3 Voorhoeve, M . 103 , 10 7

W Wagner, G . 121 , 124 Walfisz, A . 60 , 63, 81, 83 Waring's proble m 8 2 Watson, G . N . 113 , 124 Watt, N . 60-6 3 Weissbach, B . 203 , 213 Weyl, H . 1 , 15-17 , 41 , 60, 63 , 137,

149 Criterion 1 , 3 , 13-14 , 18 , 20, 27,

32, 33 , 34 differencing 17 , 40 , 42 , 46 method 39-42 , 60 , 69, 81 sum 39-46 , 65-83 , 144 , 19 6 Theorem 17 , 18 , 33

Wiener, N . 103 , 107 , 144 , 149 , 209 Wilker, J . B . 203 , 211 Wills, J . M . 203 , 213 Wintner, A . 144 , 14 9 Wirsing, E . vi , 144 , 187 , 189 , 190,

203, 209 , 21 3 polynomials 183-19 0

Wooley, T . D . vi , 82 , 8 3

Y

Yao, Q . 160 , 16 2

Z

Zaharescu, A . 37 , 19 6 zero-free regio n 163 , 168 , 169 , 17 6 Zuckerman, H . S . 19 2 Zygmund, A . 26 , 38 , 49, 64

Other Title s i n Thi s Serie s (Continued from the front of this publication)

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c

curves, 198 0

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems ,

1979

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d afEn e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finit e groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,

1977

29 A . Weinste in , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6

27 H . Blain e Lawson , Jr. , Th e quantitativ e theor y o f foliations , 197 7

26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d orders , 197 6

25 K . W . Gruenberg , Relatio n module s o f finit e groups , 197 6

24 M . Hochster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings ,

1975

23 M . E . Rudin , Lecture s o n se t theoreti c topology , 197 5

22 O . T . O'Meara , Lecture s o n linea r groups , 197 4

21 W . Stoll , Holomorphi c function s o f finite orde r i n severa l comple x variables , 197 4

20 H . Bass , Introductio n t o som e method s o f algebrai c K-theory , 197 4

19 B . Sz . -Nagy , Unitar y dilation s o f Hilber t spac e operator s an d relate d topics , 197 4

18 A . Friedman , Differentia l games , 197 4

17 L . Nirenberg , Lecture s o n linea r partia l differentia l equations , 197 3

16 J . L . Taylor , Measur e algebras , 197 3

15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z operators , 197 3

14 S . Helgason , Analysi s o n Li e group s an d homogeneou s spaces , 197 2

13 M . Rabin , Automat a o n infinit e object s an d Church' s problem , 197 2

(See th e AM S catalo g fo r earlie r titles )