THEORY OF NUMBERS

http://dx.doi.org/10.1090/pspum/008

MORGAN WARD

1901-1963

Dedicated by the Participants in the Symposium

to the Memory of

Professor Morgan Ward

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

VOLUME VIII

THEORY OF NUMBERS

American M a t h e m a t i c a l Society PROVIDENCE, RHODE ISLAND

1965

Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society

Held at the California Institute of Technology, Pasadena, California

November 21, 22, 1963

Prepared by the American Mathematical Society

under National Science Foundation Grant No. GP —1404

ALBERT LEON WHITEMAN

Editor

International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1408-7 Library of Congress Catalog Card Number65-17382

Copyright © 1965 by the American Mathematical Society Second printing, with additions, 1979

Printed in the United States of America

All rights reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers.

Contents

PREFACE vii

On the Estimation of Fourier Coefficients of Modular Forms 1 B Y ATLE SELBERG

Forms of Degrees 7 and 11 over p-adic Fields 16 B Y R. R. LAXTON AND D. J. LEWIS

An Inequality Connected with Weyl's Criterion for Uniform Distribution 22 B Y W. J. LEVEQUE

Characters and Cyclotomy 31 B Y MARSHALL HALL, J R .

Theorems on Brewer and Jacobsthal Sums. 1 44 B Y ALBERT LEON WHITEMAN

On the Degrees of the Finite Extensions of a Field 56 B Y BASIL GORDON AND E. G. STRAUS

Some Results in the Theory of Cyclotomic Fields 66 B Y KENKICHI IWASAWA

Bounds for Class Numbers 70 B Y MORRIS NEWMAN

Some New Results Connected with Matrices of Rational Integers 78 B Y E. C. DADE AND O. TAUSSKY

Essential Denominators in Normal Completions 89 B Y E. T. PARKER

The Weight of a Genus of Positive rc-ary Quadratic Forms 95 B Y GORDON PALL

Conjectures Concerning Elliptic Curves 106 B Y B. J. BIRCH

Applications of the Sieve 113 B Y N. C. ANKENY

v

VI CONTENTS

Primes Represented by Irreducible Polynomials in One Variable 119 BY PAUL T. BATEMAN AND ROGER A. HORN

Sets of Values Taken by Dirichlet's L-Series 133 B Y TOM M. APOSTOL

On the Divisor Problem 138 BY S. CHOWLA AND H. WALUM

Generating Functions and Partition Problems 144 BY L. CARLITZ

Bounded Consecutive Residues and Related Problems 170 BY W. H. MILLS

On the Additive Completion of Sets of Integers 175 BY LEO MOSER

Extremal Problems in Number Theory . 181 BY P. ERDOS

On the Shape of the Fundamental Domain of the Hilbert Modular Group 192 BY HARVEY COHN

Representations of Discrete Groups 205

BY J. LEHNER

AUTHOR INDEX 211

SUBJECT INDEX 213

Preface The six hundred sixth meeting of the American Mathematical Society

was held on November 21-23, 1963 at the California Institute of Technology in Pasadena, California. By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, and with the financial support of the National Science Foundation, a Symposium on Recent Developments in the Theory of Numbers was held on November 21 and 22 in conjunction with this meeting. The Program Committee for the Symposium consisted of Professors Leonard Carlitz, D. H. Lehmer, W. J. LeVeque, and A. L. Whiteman, Chairman.

The following pages contain all but two of the twenty-four papers which were presented at the Symposium. The paper, On the lack of unique factor­ization in quadratic fields, presented by Professor Ivan Niven and Professor Herbert S. Zuckerman, will be published elsewhere. The paper. On p-adic analysis, delivered by Professer Bernard M. Dwork, will appear in the Annals of Mathematics under the title On the zeta function of a hypersur-face II .

The purpose of the Symposium was to bring together experts in the allied branches of the broad field of number theory. By fostering open discussions the Symposium proved to be a vigorous and highly successful means of disseminating information about the latest advances in mathe­matical knowledge. The interaction amongst the participants has unques­tionably served as an impetus for future important research.

There were four sessions of the Symposium each preceded by an hour long principal address. The sessions consisted of fifteen minute talks and were designated as follows: I. Session on Diophantine Analysis and Alge­braic Number Theory; II. Session on Matrices and Quadratic Forms: III . Session on Analytic Number Theory; and IV. Session on Analytic Number Theory and Modular Functions. During the second session Professor D. H. Lehmer gave a brief talk on the work of the late Morgan Ward in number theory. The four addresses were given by Professors Selberg, Iwasawa, Birch and Carlitz. The order of the papers in these Proceedings is the same as the order in which the talks were presented.

This volume is dedicated to the memory of Professor Morgan Ward. His untimely death on June 26, 1963 kept him from delivering one of the invited lectures.

Albert Leon Whiteman University of Southern California

V l l

Author Index

Italic numbers refer to pages on which a complete reference to a work is given. Roman numbers refer to pages on which a reference is made to a work of the author. For

example, under Ward would be the page on which a statement like the following occurs: "This theorem was proved by Ward [6, p. 45] in the following manner • • • ."

Boldface numbers indicate the first page of articles in this volume.

Albert, A. A., 90, 94 Ankeny, N. N., 70, 77, 113 Apostol, T. M., 133,134,137 Artin, E., 58, 63, 64, 65, 66, 67

Bachmann, P., 47, 48, 55 Bateman, P. T., 119, 120, 121, 122, 123,

128,131 Behrend, F. A., 181,189 Bell, E. T., 156,168 Bellman, R., 146,168 Bianchi, L., 190, 202 Bierstedt, R. G., 170,174 Birch, B. J., 16, 17,19, 21, 106, 109, 111 Blumenthal, O., 190, 202 Bohr, H., 133,137 Borel,A.,108, 111 Bourbaki, N., 57, 65 Brauer, Alfred, 170,174 Brewer, B. W., 44, 45, 55 Brillhart,J. D., 170,174 Bruck, R. H., 89, 91, 94 deBruijn, N. G., 175, 180 Burnside, W., 34, 42

Carlitz, L., 144, 146, 148, 156, 159, 160, 163, 168

Cassels, J. W. S., 106,108, 109,110, 111,222

Cauchy, A., 45, 55 Chaundy, T. W., 146,160,168 Cheema, M. S., 146,148,168 Chowla, S., 70, 77, 138 Cohn, Harvey, 190,190, 202

Dickson, L. E., 31, 35, 42, 53, 55 Dunton, M., 170,174 Dwork, B., 107,112

Eichler, M., 4,14 Eisenstein, G., 44, 55 Erdos, P., 22, 30,175, 176,180, 181, 181,

182,188,189 Euler, L., 91, 94

Faddeev, D. K., 83, 88 Feit, Walter, 34, 42 Fine, N. J., 145,168 Forsyth, A. R., 146, 168 Fricke, R., I l l , 112, 190, 191, 202

Goldhaber, J. K., 89, 90, 91, 94 Gordon, Basil, 56,146, 147, 163,168, 169 Gotzky, F., 191, 202 Graham, R .L . , 170,174 Gut, M., 70, 77

Hall, Marshall, Jr., 31, 34, 42, 43, 89, 90, 94 Hardy, G. H., 3, 15, 119, 120, 130, 131, 146,

147, 169 Hasse, H., 16, 21, 37, 42, 67,107, 112 Hecke, E., 4, 15, 111,222 Horn, R. A., 119, 120,122, 123, 128,131 Hua, L. K., 75, 77

Iwasawa, Kenkichi, 66

Jacobsthal, E., 49, 55 Johnsen, E. C., 89, 91, 94

Dade, E. C., 78, 78, 88 Davenport, H., 3,14,16, 17, 19, 21, 37, 42,

107,112 Demjanov, V. B., 16, 21 Deuring, M., 107, 110,112 DeVore,R. , 191, 199,202

Klein, F., 190, 191, 202 Kloosterman, H. D., 3,15 Koksma, J. F., 22, 29, 30, 195, 202

Landau, E., 50, 55,14, 77 Lang, S., 16, 17, 21, 106, 108, 109, 111, 112

211

212

Laxton, R. R., 16 Lehmer, D. H., 170, 173, 174 Lehmer, Emma, 127, 131, 170, 173, 174 Lehner, J. ,203 Leveque, W. J., 22 Lewis, D. J., 16,16, 17, 19,27 Linnik, Ju. V.,9, 15 Littlewood, J. E., 119, 120, 130, 131 Lukacs, E., 23, 30 Lutz, E., 106, 112

Maass, H., 190, 202 Macauley, F. S., 17, 21 Macbeath, A. ML, 205, 208 MacMahon, P. A., 145, 146, 165, 169 MacNeish, H. F„ 93, 94 Mann, H. B., 91, 93, 94 Miech, R. J., 121, 131, 131 Mills, W. H., 170, 170, 171, 174 Minkowski, H., 73, 77, 84, 88, 95, 105 Mitchell, H. H., 31, 38, 43 Mordell,L.J., 106, 112 Moriya, M., 61, 65 Moser, Leo, 175, 188, 189 Murphy, T. G., 16, 21

Nagell, T., 106, 112 Narkiewcz, W., 176, 180 Newman, Morris, 70

Ono,T., 108, 110, 112

Paley, R. E. A. C , 39, 43 Pall, Gordon, 95, 96, 97, 102, 105 Parker, E. T., 89 Peterson, H., 3, 15

Rankin, R. A., 3, 15 Renyi,A., 131, 131 Riordan, John, 149, 155, 157, 163, 168, Robinson, D. W., 78, 88

AUTHOR INDEX

Roth, K. F., 29, 30, 181,189 Ryser ,H.J . ,89 , 90, 94

Salie, H., 3, 15 Schaffstein, K., 76, 77 Schinzel, A., 119, 130, 131 Schreier, O., 58, 63, 64, 65 Schur, I., 188, 189 Selberg, A., 1, 3, 4, 5, 7, 10, 15, 121, 132 Selfridge,J. L., 170, 174 Selmer, E. S., 109, 112 Serre,J.-P., I l l , 112 Shanks, Daniel, 122, 132 Shimura, G., 111,112 Siegel, C. L., 95, 96, 105, 190, 202 Sierpihski, W., 119, 131 Smith, H. J. S., 95,105 Sperner, E., 185, 189 Springer, T. A., 16, 19, 21 Stemmler, M., 121, 122, 131 Stern, M., 44, 55 Straus, E. G., 56 Swinnerton-Dyer, H. P. F., 109, 111

Tate, J., 66, 108, 109, 111, 112 Taussky, O., 78, 78, 83, 87, 88 Thompson, J. G., 34,42 Touchard, J., 149, 169 Turan, P., 22, 30, 134, 136, 137, 181, 189

Vandiver, H. S., 31, 43

Walum, H., 138 Wang, Yuan, 121, 122,132 Ward, M., 78, 88 Weber, H., 50, 55 Weil, A., 3, 15, 16, 17, 21, 106, 107, 108, 112 Weyl, H.,22,30, 70, 77 Whiteman, A. L., 31, 43, 44, 44, 47, 48, 49, 55 Witt, E., 97, 105

169 Wrench, J. W. Jr., 122, 132 Wright, E. M., 146, 147, 169

Subject Index

Abelian polynomial, 122 Algebra, order in an, 79 Arithmetic type, 2 Arrays

two line, 158 weighted, 166

Artin and Schreier, theorem of, 58 Automorphic set, 97

B-constructible, 58 Bell

numbers, 156 polynomial, 155

Bernoulli polynomial, 138 Bessel function, 8 Bipartite, 146 Bodies, convex, 196 Bohr's theorem, 133, 134, 136 Bounds

for h for quadratic fields, 74 upper for h, 70

Canonical residue(s), 99 unique, 99

CE field, 61 Cebysev polynomial, 157 Circle method, 3

Hardy-Littlewood's, 3 Circulants, integral, 83 Class(es),95

number, 73, 74 Conpact, 204, 205, 207

group, 206 discrete, 204

Companion matrix, 78 Complement, 96 Completability, 92 Congruence-subgroup(s), 5, 13 Congruent modulo, 95 Conjecture, Ramanujan, 2 Consecutive residues, 170 Constants, cyclotomic, 31 Convex bodies, 196 Covering system, 182 Criterion, Weyl's, 22 Curve, zeta function of elliptic, 107 Cusp form(s), 1, 2

Fourier coefficients of, 13

representation of, 3 vector, 6

Cyclotomic constants, 31 number, 47

Cyclotomy, theory of, 45, 47

Degenerate form, 16 Degree(s)

set, 56 of normal, 56

Denominator, essential, 91 Density of the genus, p-adic, 96 Derivative, 66 Dickson-Hurwitz sums, 47, 48 Difference sets, 31 Differentials, 190 Dirichlet series, 4, 9

equivalent, 134 Discontinuous, 204 Discrepancy, 22 Discrete group(s), 203

compact, 204 noncompact, 204 representations of, 203

Disjoint, 182 Distribution function, 22 Divisor

of a set, 59 problem, 138

Eigenfunction(s), 10, 12 orthonormal system of, 10

Eigenvalues, 10 Elliptic

curve, zeta function of, 107 fixed point, 197

Epstein zeta function, 5 Equation(s)

incidence, 89 normal, 90

Equivalent Dirichlet series, 134 forms, unimodular, 17

E . R . H . , 116 Essential denominator, 91 Exponential sums, 3, 22 Extended Riemann Hypothesis, 115

213

214 SUBJECT INDEX

/(»), 181 Fermat, 191 F-group(s), 203, 204, 205, 207 Field(s)

CE,61 finite, 166 local cyclotomic, 66 p-adic, 16 quadratic, bounds for h for, 74

Finite field, 166 Fourier series, 47 projective planes, 89

Fixed points, 197 elliptic, 197

Floor(s),194, 196 nonsimple, 199

Form(s) cusp, 1

representation of, 3 Fourier coefficients of, 13

degenerate, 16 Hermite normal, 71 modular

of type ( — k,x, r"), 1 Fourier coefficients of, 1

reduced, 17 unimodular equivalent, 17 vector, 6 weight of a, 95

Fourier coefficients of

cusp forms, 13 modular forms, 1

series, finite, 47 Free, 206, 207

group,204, 205, 207 product, 205, 206, 207, 208

Function(s) Bessel, 8 distribution, 22 Epstein zeta, 5 group ring, 35 Ramanujan, 2 Riemann zeta, 142 zeta, 4

Fundamental group,204, 205,206 region, 204, 205

£(n),181 General sieve, 113

Genus, 95, 204, 206 p-adic density of, 96 weight of a, 95

Goldbach conjecture, 131 Group(s)

compact, 206 discrete, 204

F-, 203, 204 free, 204, 205, 206 fundamental, 204, 205, 206 modular, 203, 204 noncompact, 206

discrete, 204 representations of discrete, 203 ring functions, 35 Tate-Safarevic, 108

Hardy-Littlewood's circle method, 3 Hecke operators, 4 Hermite normal form, 71 Hilbert

fundamental domain, 190 norm residue symbol, 66

Hypothesis H, 119 L, 130

Incidence equation(s), 89 normal, 90

Integral circulants, 83 matrix, 89 -valued polynomials, 128

ip type, 99

Jacobi sum, 47, 50 theta formula, 147

Jacobsthal sum, 44, 48, 54 theorem of, 49

Kloosterman sums, 3, 7, 8 Kolataroff, 195 Korkine, 195 Kronecker symbol, 8

A(k,m), 170 Latin square, 92 Lattice permutations, two element, 165 Local cyclotomic field, 66 Low points, 198

SUBJECT INDEX 215

Matrices, rational, 89 Matrix

companion, 78 integral, 89 normal, 83 residue, 84

Means, method of, 176 Method

of means, 176 variance, 177

Modular form(s)

Fourier coefficients of, 1 o f t y p e ( - * , x , r ' ) , l

group, 203, 204 Modulo, congruent, 95 Multipartite, 146 Multiplier, 1, 12

system, 1, 12

Point(s) elliptic fixed, 197 fixed, 197 low, 198

Polynomial abelian, 122 Bell, 155 Bernoulli, 138 Cebysev, 157 integral-valued, 128 nonabelian, 122

pp type, 99 Presentation(s), 203, 204, 205, 207 Prime(s)

number theorem, 119 twin,130

Primitive, 96 Principal order, 78 Product, free, 205 Projective planes, finite, 89

Nonabelian polynomial, 122 Noncompact, 204, 205, 207

discrete group, 204 group, 206

Nonsimple floors, 199 Normal

form, Hermite, 71 incidence equations, 90 matrix, 83

Number (s) Bell, 156 class, 73, 74 cyclotomic, 47 of first descents, 110 Tamagawa, 108 theorem, prime, 119

Operators, Hecke, 4 Order

in an algebra, 79 principal, 78

Orthonormal system of eigenfunctions, 10

p-adic density of the genus, 96 p-adic field, 16 Partition(s), 144

plane, 145 unrestricted, 144

Plane(s) finite projective, 89 partitions, 145

Quadratic fields, bounds for h for, 74

rk(n), 1SI Ramanujan

conjecture, 2, 14 function, 2

Rankin's method, 4, 14 Ratio, Tamagawa, 110, 111 Rational matrices, 89 Reduced forms, 17 Region, fundamental, 204 Representation(s), 97, 203, 207

of cusp forms, 3 of discrete groups, 203

Represented, 97 Residue(s)

canonical, 99 unique, 99

consecutive, 170 matrix, 84 symbol, Hilbert's norm, 66 unique canonical, 99

Riemann hypothesis, extended, 115 surface, 204, 205 zeta function, 142

Ring functions, group, 35

Schreier, and Artin, theorem of, 58 Section, 101 Segment, 101

216 SUBJECT INDEX

Seimer conjecture, 108, 109, 110 Series

Dirichlet, 4, 9 equivalent, 134

finite Fourier, 47 Set(s)

automorphic, 97 degree, 56 difference, 31 divisor of, 59 of normal degrees, 56

Siegel, 190 Sieve

general, 113 method, 113, 121

Signature, 203, 204 Simple, 196 Square

latin, 92 roots of Z-matrices, 79

Sum(s) Dickson-Hurwitz, 47, 48 exponential, 3, 22 Jacobi, 47, 50 Jacobsthal, 44, 48, 54 Kloosterman, 3, 7, 8

Surface, Riemann, 204 Symbol, Kronecker, 8 System

covering, 182 multiplier, 1, 12

Tableaux, Young, 163 Tamagawa

number, 108 ration, 110, 111

Tate-Safarevic conjecture, 109

group, 108 Theorem of

Artin and Schreier, 58 Jacobsthal, 49

Theory of cyclotomy, 45, 47 Titchmarsh, 142 Twin primes, 130 Two

element lattice permutations, 165 line arrays, 158

weighted, 166

Uniformly distributed (mod 1), 22 Unimodular, 95, 97

equivalent forms, 17 Unique canonical residue, 99 Unrestricted partitions, 144 Upper bounds for h, 70

Valence, 191, 194 Variance method, 177 Vector cusp form, 6 Voronoi, 139

Weight of a form, 95 genus, 95

Weighted two line arrays, 166 Weyl's criterion, 22 Witt-type theorems, 97

Young tableaux, 163

Zeta function, 4 Epstein, 5 of an elliptic curve, 107

Z-matrices, 78 square roots of, 79