'

CONTEMPORARY . MATHEMATICS

166

The Rademacher Legacy to Mathematics

The Centenary Conference in Honor of Hans Rademacher

July 21-25, 1992 The Pennsylvania State University

George E. Andrews David M. Bressoud

L. Alayne Parson Editors

Recent Titles in This Series

166 George E. Andrews, David M. Bressoud, and L. Alayne Parson, Editors, The Rademacher legacy to mathematics, 1994

165 Barry Mazur and Glenn Stevens, Editors, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, 1994

164 Cameron Gordon, Yoav Moriah, and Bronislaw Wajnryb, Editors, Geometric topology, 1994

163 Zhong-Ci Shi and Chung-Chun Yang, Editors, Computational mathematics in China, 1994

162 Ciro Ciliberto, E. Laura Livorni, and Andrew J. Sommese, Editors, Classification of algebraic varieties, 1994

161 Paul A. Schweitzer, S. J., Steven Hurder, Nathan Moreira dos Santos, and Jose Luis Arrant, Editors, Differential topology, foliations, and group actions, 1994

160 Niky Kamran and Peter J. Olver, Editors, Lie algebras, cohomology, and new applications to quantum mechanics, 1994

159 William J. Heinzer, Craig L. Huneke, and Judith D. Sally, Editors, Commutative algebra: Syzygies, multiplicities, and birational algebra, 1994

158 Eric M. Friedlander and Mark E. Mahowald, Editors, Topology and representation theory, 1994

157 Alfio Quarteroni, Jacques Periaux, Yuri A. Kuznetsov, and Olof B. Widlund, Editors, Domain decomposition methods in science and engineering, 1994

156 Steven R. Givant, The structure of relation algebras generated by relativizations, 1994 155 William B. Jacob, Tsit-Yuen Lam, and Robert 0. Robson, Editors, Recent advances in

real algebraic geometry and quadratic forms, 1994 154 Michael Eastwood, Joseph Wolf, and Roger Zierau, Editors, The Penrose transform and

analytic cohomology in representation theory, 1993 153 RichardS. Elman, Murray M. Schacher, and V. S. Varadarajan, Editors, Linear algebraic

groups and their representations, 1993 152 Christopher K. McCord, Editor, Nielsen theory and dynamical systems, 1993 151 Matatyahu Rubin, The reconstruction of trees from their automorphism groups, 199 3 150 Carl-Friedrich Biidigheimer and Richard M. Hain, Editors, Mapping class groups and

moduli spaces of Riemann surfaces, 1993 149 Harry Cohn, Editor, Doeblin and modern probability, 199 3 148 Jeffrey Fox and Peter Haskell, Editors, Index theory and operator algebras, 1993 147 Neil Robertson and Paul Seymour, Editors, Graph structure theory, 1993 146 Martin C. Tangora, Editor, Algebraic topology, 1993 145 Jeffrey Adams, Rebecca Herb, Stephen Kudla, Jian-Shu Li, Ron Lipsman, and Jonathan

Rosenberg, Editors, Representation theory of groups and algebras, 1993 144 Bor-Luh Lin and William B. Johnson, Editors, Banach spaces, 199 3 143 Marvin Knopp and Mark Sheingorn, Editors, A tribute to Emil Grosswald: Number

theory and related analysis, 1993 142 Chung-Chun Yang and Sheng Gong, Editors, Several complex variables in China, 1993 141 A. Y. Cheer and C. P. van Dam, Editors, Fluid dynamics in biology, 1993 140 Eric L. Grinberg, Editor, Geometric analysis, 1992 139 Vinay Deodhar, Editor, Kazhdan-Lusztig theory and related topics, 1992 138 Donald St. P. Richards, Editor, Hypergeometric functions on domains of positivity, Jack

polynomials, and applications, 1992 137 Alexander Nagel and Edgar Lee Stout, Editors, The Madison symposium on complex

analysis, 1992 (Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/166

The Rademacher Legacy to Mathematics

Hans Rademacher

CoNTEMPORARY MATHEMATICS

166

The Rademacher Legacy to Mathematics

The Centenary Conference in Honor of Hans Rademacher

July 21-25, 1992 The Pennsylvania State University

George E. Andrews David M. Bressoud

L. Alayne Parson Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL BOARD Craig Huneke, managing editor

Clark Robinson J. T. Stafford Linda Preiss Rothschild Peter M. Winkler

The Hans Rademacher Centenary Conference was held at The Pennsylvania State University, from July 21 to July 25, 1992, with support from the National Science Foundation, Grant No. DMS 9123821; and the National Security Agency, Grant No. MDA904-92-H-3055; and the Institute for Mathematics and its Appli-cations.

1991 Mathematics Subject Classification. Primary llFxx; Secondary 11A55, llGxx, llLxx, llPxx.

Library of Congress Cataloging-in-Publication Data The Rademacher legacy to mathematics: The centenary conference in honor of Hans Rademacher, July 21-25, 1992, the Pennsylvania State University/ George E. Andrews, David M. Bressoud, L. Alayne Parson, editors.

p. em. -(Contemporary mathematics, ISSN 0271-4132; v. 166) "[Proceedings of] the Hans Rademacher Centenary Conference"-Pref. Includes bibliographical references. ISBN 0-8218-5173-X 1. Automorphic forms-Congresses. 2. Discontinuous functions-Congresses. I. Rademacher,

Hans, 1892-1969. II. Andrews, George E., 1938-. III. Bressoud, David M., 1950-. IV. Par-son, L. Alayne, 1947-. V. Hans Rademacher Centenary Conference (1992: Pennsylvania State University) VI. Series: Contemporary mathematics (American Mathematical Society); v. 166. QA331.R33 1994 94-12052 512'.73-dc20 CIP

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Contents

Preface George E. Andrews

Hans Rademacher 1892-1969 Bruce C. Berndt

On Rademacher's multiplier system for the classical theta-function Bruce C. Berndt and Ronald J. Evans

Application of Dedekind eta-multipliers to modular equations Harvey Cohn and Marvin I. Knopp

Singularities, functional equations, and the circle method Leon Ehrenpreis

On Weyl's inequality for eigenvalues Joseph Lehner and G. Milton Wing

The GL(2) Rankin-Selberg convolution for higher level non-cuspidal forms Daniel B. Lieman

Congruences on the Fourier coefficients of modular forms on r 0 (N) Ken Ono

Diagonalizing Eisenstein series IV Robert A. Rankin

Generalized Dedekind ry-products Sinai Robins

vii

xi

xiii

1

9

35

81

83

93

107

119

viii CONTENTS

Vanishing coefficients in the expansion of products of Rogers-Ramanujan type Krishnaswami Alladi and Basil Gordon

Schur's theorem, Capparelli's conjecture and q-trinomial coefficients George E. Andrews

Modified convergence for q-continued fractions defined by functional relations

129

141

Douglas Bowman 155

A simple proof of an Aomoto type extension of the q-Morris theorem Kevin W. J. Kadell 167

Congruences for certain generalized Frobenius partitions modulo p3

Louis Worthy K olitsch 183

Some asymptotic formulae for Ramanujan's mock theta functions Richard J. Mcintosh 189

On the distribution of Dedekind sums Roelof W. Bruggeman 197

Wilf's conjecture and a generalization Gert Almkvist 211

An elementary proof of Wilf's conjecture Tim Dokshitzer 235

Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration Frank Garvan 245

Residue periodicity in subgroup counting functions Michael Grady and Morris Newman 265

Addition theorems for a-finite groups Xing-De Jia and Melvyn B. Nathanson 275

On sums of three integral cubes W. Conn and L. N. Vaserstein 285

A note on character sums J. B. Friedlander and H. lwaniec 295

Large factors of small polynomials David W. Boyd

CONTENTS

Factors of period polynomials for finite fields, I S. Gurak

The Phragmen-Lindelof theorem and modular elliptic curves Liem Mai and M. Ram Murty

Multidimensional beta and gamma integrals Ronald J. Evans

Rademacher, calculus of variations, inequalities, and relative error Kenneth B. Stolarsky

ix

301

309

335

341

359

Preface

From July 21 to July 25, 1992 the Hans Rademacher Centenary Conference was held at the Pennsylvania State University. Hans Rademacher was born in 1892, and his life is well described in the biography by Bruce Berndt which was originally published in Acta Arithmetica and has been updated for this volume.

The organizers of the conference were Bruce Berndt, David Bressoud, Marvin Knopp, Herb Wilf and myself. Each of these individuals was immensely sup-portive of this tribute to Hans Rademacher, and their love of his mathematical tradition and his legacy moved them to assist in making the conference a great success.

Alayne Parson, David Bressoud and I edited this volume. It is only just to point out that Alayne and David did the great majority of the work in getting this volume to press in collaboration with Donna Harmon at the A.M.S. Also special thanks should be given to Kathy Wyland who typed and/or retyped a number of papers.

It is an understatement to suggest that Rademacher's mathematical interests were broad, and we tried to represent these topics in the papers presented at the conference. The theory of modular forms and recent developments are naturally a significant portion of this volume (Berndt & Evans, Cohn & Knopp, Ehren-preis, Lehner & Wing, Lieman, Ono, Rankin, Robins). Closely related are the papers in partitions and q-series ( Alladi & Gordon, Andrews, Bowman, Kadell, Kolitsch, Mcintosh). The papers by Bruggeman (on Dedekind Sums), Almkvist and Dokshitzer (on a conjecture by Herb Wilf made at the beginning of the conference and also solved by A. 0. L. Atkin), and Garvan (on Ramanujan type identities and the A. G. M.) all closely relate to the first two topics.

Of course there has always been a significant group-theoretic component con-nected with modular forms and this area is represented by the papers of Grady & Newman and Jia & Nathanson.

Other topics of interest to Rademacher in number theory are covered by Conn & Vaserstein, Friedlander & Iwaniec, Gurak and Mai & Murty.

Pure analysis describes Evans' paper, and Stolarsky's paper is concerned with Rademacher's last paper on "the roundest oval."

xi

xii PREFACE

On Wednesday evening July 22, 1992 we held an evening event with the title Remembering Rademacher at which Marvin Knopp, Al Whiteman (via letter), Paul Bateman, Joe Lehner, Morris Newman, George Andrews, Karin Rademacher Loewy, Herb Wilf, Ray Ayoub, Ken Stolarsky and Kevin Kadell spoke. Each of us attending that event learned much about Rademacher that we had not known before. Especially memorable was the tribute from his daughter Karin Rademacher Loewy. Of the remarks made by his students, I think the one that struck a chord in each of us was the story told by Paul Bateman:

"The help that Rademacher gave me with respect to my thesis con-sisted of the following: He suggested the topic and he suggested that I might find it helpful to use the Poisson Summation Formula. On one occasion when I asked Hans for his advice on getting out of a log jam, he uttered a few words that ... were as follows "Young man, it's your dissertation." Now I should emphasize that he did not say this in a negative way. He said this in a way that indicated that he knew very well that I could do it myself and he wanted me to believe that too."

Others there that evening alluded to this story. It probably struck no one more than me. I vividly remember Rademacher proposing that I study Ramanujan's mock theta functions and that the Poisson Summation Formula might be helpful.

It is thus with great pleasure that I commend to you this proceedings of the Hans Rademacher Centenary Conference.

George E. Andrews University Park, Pennsylvania February 15, 1994

Contemporary Mathematics Volume 166, 1994

Hans Rademacher 1892-1969

BY

BRUCE C. BERNDT

Hans Rademacher was born of Lutheran parents on April 3, 1892 in Wands-bek, near Hamburg. His mother and father, Emma and Henry, owned a local store. There were two others in the family, a brother Martin and a sister Erna.

By the time Rademacher entered the University in Gottingen in 1910, he had developed a great breadth of interests, an attribute that would be transformed in his later mathematical life into a creative interest in many branches of mathe-matics. At the age of 18 he had a curiosity for mathematics, the natural sciences, foreign languages, and philosophy. It was to the latter area that Rademacher devoted his primary initial attention. However, he enjoyed the lectures of two of Felix Klein's assistants, Erich Heeke and Hermann Weyl, and eventually he turned to mathematics, chiefly through the influence of Richard Courant. His doctoral dissertation [1 J was in real analysis and was completed under the direc-tion of C. Caratheodory in 1916, despite the fact that for portions of the years 1914-1916 he served with his country's army.

Rademacher's first position was as a teacher in private secondary schools in Bischofstein and Wickersdorf. From 1919-1922 he served as a Privat Dozent in Berlin. In 1922 he accepted the position of Ausserordentlicher Professor at the University of Hamburg, a position he held until being named Professor at Breslau in 1925. While in Berlin, he married Susanna Gaspary, and their daughter Karin was born in 1925. The marriage ended in divorce in 1929.

Throughout his entire life, Rademacher maintained an active interest in hu-man rights. In Breslau he joined the International League for the Rights of Men and was chairman of the local chapter of the Deutsche Friedensgesellschaft. Not

1991 Mathematics Subject Classification. Primary 11-03, 01A70; Secondary 11F03, 01A60. This paper is in final form and no version of it will be submitted for publication elsewhere.

xiii

© 1994 American Mathematical Society 0271-4132/94 $1.00 + $.25 per page

xiv BRUCE C. BERNDT

surprisingly, after Hitler's ascendancy to power, Rademacher was removed from his position in 1934. He and his daughter Karin moved briefly to the small vil-lage, Niehagen, on the Baltic Sea where he married Olga Frey. In October of that same year, Rademacher arrived in the United States, where he had been invited to the University of Pennsylvania as a Visiting Rockefeller Fellow. Rademacher's wife stayed behind in Germany but came to the United States in January, 1935. Their son Peter was born in November, 1935. Upon the completion of his fellow-ship, Rademacher returned briefly to Germany but came back in the autumn of 1935 to accept a position at the University of Pennsylvania, the institution he was to serve faithfully until his retirement in 1962. His daughter Karin stayed behind in Germany, and finally in 1947 he succeeded in helping her come to the United States.

Although Rademacher had held a full professorship in Germany for ten years before immigrating to the United States, he was offered only an assistant pro-fessorship at Pennsylvania in 1935. Despite this, Rademacher was ever after to remain loyal to the University of Pennsylvania for providing him refuge from the horror that had engulfed his native land. Rademacher enjoyed congenial relations with many colleagues and graduate students at the University of Pennsylvania as well as in Swarthmore where he and his family resided. Among his best friends were A. Zygmund, J. R. Kline, A. Dresden from Swarthmore College, and I. J. Schoenberg, whom he had previously known in Germany (and who recently died on February 21, 1990 at the age of 86). Rademacher's second marriage also ended in divorce in 1947, and he moved to Philadelphia close to the campus of the University of Pennsylvania. In 1949, Rademacher married Schoenberg's sister Irma Wolpe, who had a successful career as a concert pianist. She and her husband opened their home frequently to mathematicians, musicians, and a host of other friends. An annual event at the Rademacher home was an informal New Year's Day afternoon gathering.

During his residence in Swarthmore, Rademacher's good friend A. Dresden introduced him to the Society of Friends. Rademacher joined the Society and remained a member until his death.

In Germany and while at Pennsylvania, Rademacher supervised the Ph.D. dissertations of twenty-one students, many of whom became well-known mathe-maticians. These twenty-one doctoral students were Theodor Estermann, Wolf-gang Cramer, Otto Schulz, Kathe Silberberg, Albert Whiteman, Joseph Lehner, Lowell Schoenfeld, Ruth Goodman, John Livingood, Paul Bateman, Jean Wal-ton, Nelson Brigham, Emil Grosswald, Saul Rosen, Leila Dragonette, Albert Schild, Jean Calloway, Morris Newman, Frederick Homan, William Spohn, and George Andrews. At the time of this writing, the Rademacher mathematical family tree is blooming into its fifth generation with over 175 flowers.

In 1938, Rademacher and James Clarkson began to conduct a problems sem-inar at Pennsylvania. Later, after Clarkson departed for service in World War II, Schoenberg took his place. Through the seminar's success over several years,

HANS RADEMACHER XV

many of the problems became widely known. The seminar was officially listed in the University of Pennsylvania's catalog as the "Proseminar." It was a re-quired course for students seeking a graduate degree, and was often used in the determination of a student's progress and the awarding of fellowships.

During the spring semester of 1953, Rademacher held a position at the Insti-tute for Advanced Study in Princeton. In 1954-55, he lectured at the Tata Insti-tute of Fundamental Research in Bombay and at the University of Gottingen. In 1952 and 1959-60, he was Phillips Lecturer at Haverford College, and in 1960-61 he again visited the Institute for Advanced Study. He also had lengthy stays at the University of Oregon and at the University of California at both Berkeley and Los Angeles.

During his final year at Pennsylvania, a special year in the theory of numbers was convened, and several prominent number theorists participated in this trib-ute to Rademacher. His university awarded him an honorary doctorate upon his retirement in 1962. After retiring from the University of Pennsylvania, Rademacher lectured at New York University for two years. In 1964, he ac-cepted a position at Rockefeller University in New York, a post he held until his death in Bryn Mawr on February 7, 1969. Rademacher became a founding editor of Acta Arithmetica in 1935 and served on its editorial board until his death.

Rademacher made significant contributions to several areas of mathematics, including real analysis and measure theory, complex analysis, number theory, geometry, and numerical analysis. He also coauthored two papers in physics (21], (22] and wrote an article in genetics (31]. About 50 of the 76 papers listed in his bibliography are concerned with number theory or related areas. In this brief survey of Rademacher's achievements, we shall confine ourselves mainly to these 50 articles. Note that there are two bibliographies at the end of this essay. The second comprises the works of Rademacher divided into three categories: papers, books and lecture notes, and problems; the first bibliography contains articles of other authors cited in this survey.

The fourteen papers published by Rademacher in 1916-1922 chiefly concern the theory of functions of a real variable and measure theory. In particular, we mention paper (13] in which Rademacher introduced an orthogonal system of functions now known as Rademacher functions. He soon wrote a sequel on the completion of this system but was discouraged from publishing it. In the meanwhile, J. L. Walsh (64] independently completed the system and published a paper containing much of Rademacher's work. Rademacher never published his paper, which, for many years, was considered to be permanently lost. In the summer of 1979, the paper was miraculously unearthed, and soon thereafter E. Grosswald (19] described the paper's history and its contents at a confer-ence honoring Grosswald at his retirement. Since its discovery, Rademacher's orthonormal system has been utilized in numerous instances in many areas of analysis.

xvi BRUCE C. BERNDT

Papers [3] and [10] might be classified under the purview of number theory as well as analysis. In [3], Rademacher gives an easy proof of a theorem of Borel: The probability that in a decimal expansion not all ten digits appear asymptotically the same number of times is zero. A sharper version of the theorem is proved in [10]. If nv(x) denotes the number of times the digit v, 0 :::; v :::; 9, appears in the first n digits of the decimal expansion of a number x E (0, 1), then for any f > 0,

as n tends to oo, except on a set of measure 0. Actually, a stronger theorem was previously given by G. H. Hardy and J. E. Littlewood [22], [21, pp. 28-63]. Rademacher's proof, however, is short and elegant. It depends upon one of his theorems giving necessary and/ or sufficient conditions on the sequence {en} of complex numbers for the convergence of I: unen, where I: Un is divergent and Un is strictly decreasing to 0.

Rademacher's first particularly notable contribution to number theory is his improvement of Brun's sieve [14]. As an application of his theorem, Rademacher showed that if f(x) is an irreducible, primitive polynomial of degree g, these-quence {! ( n)} contains infinitely many terms with at most 4g - 1 prime factors, not counting those which appear in every term of the sequence. For the twin prime problem, it follows from Rademacher's work that there exist infinitely many pairs n, n + 2 such that each member contains no more than seven prime factors. From Brun's original theorem, this conclusion could be drawn if "seven" were replaced by "nine." In [15], Rademacher showed how Brun's method could be applied to algebraic number fields. For a lucid, contemporary account of Brun's sieve, see Halberstam and Richert's treatise [20, pp. 56-68].

In [16], [17], and [23], Rademacher generalized to algebraic number fields the work of Hardy and Littlewood [23], [21, pp. 561-630] on expressing a positive integer as a sum of three or more primes. Hardy and Littlewood had shown that if there is a number (} < 3/4 such that no Dirichlet £-function has a zero with real part greater than (}, then, subject to an obvious parity condition, there is an asymptotic formula for the number of ways of expressing a positive integer as a sum of m primes, where m ~ 3. Rademacher's extension of this result to algebraic number fields requires the existence of a number(}' < 3/4 such that no Heeke zeta-function for the relevant algebraic number field has a zero with real part greater than(}'. Just as the results of Hardy and Littlewood [23] were later established by I. M. Vinogradov and A. Walfisz with no unproved hypothesis, in an analogous way, Rademacher's papers [16], [17], and [23] were later superseded by the work of 0. Korner [32], [33], [34].

The majority of Rademacher's subsequent contributions to number theory came in the related areas of modular forms, partition problems, and Dedekind sums. We first describe a few miscellaneous results.

HANS RADEMACHER xvii

Rademacher's first research paper in the theory of modular forms is [26]. Us-ing the Schreier-Reidemeister method, he determined a system of generators for the congruence subgroups ro(P) of the modular group, where pis a prime. He also determined all relations involving the generators. Later, H. Frasch (15] and Grosswald (17] proved similar theorems for the groups r(p) and rg(p), respec-tively. Grosswald (18] also proved that under certain conditions the generators for r(p) could be chosen so that none is parabolic. It was not until 1973 that corresponding theorems for r 0 (N), where N is an arbitrary positive integer, were established by Y. Chuman (12]. A completely new method, based on Farey symbols, has recently been devised by R.S. Kulkarni (35].

In (65] and (68], Rademacher gives a short, clever proof, based on a corollary of Jensen's inequality, of the following well-known theorem: A modular function analytic and bounded on the open upper half-plane and belonging to a congru-ence subgroup of the modular group is necessarily a constant.

Recall that the Dedekind eta-function 17( T) is defined for Im T > 0 by 00

17(T) = e'lrir/12 II {1- e21rinr).

n=l

Rademacher devoted several papers to the study of this function and the Dedekind sums which appear in the transformation formulae of log 17( T). In order to define the Dedekind sum s{ h, k), set

Then

{ x - (x] - 1/2, ((x)) =

0,

k

if x is not an integer, otherwise.

s(h, k) := "f)(hJ.L/k))((J.L/k)), Jl=l

where h and k are coprime integers with k 2': 1. Rademacher's first paper devoted to the eta-function and Dedekind sums was

published in 1931 (27]. Here he proves the transformation formula for log17(r) under modular transformations via contour integration and the functional equa-tion of the Hurwitz zeta function. A similar proof of the transformation formula for T(r) = -1/r was later given by A. Weil (66]. One of the most beautiful methods for obtaining the transformation formulae for log 17( T) is by contour integration and is due to C. L. Siegel (61) and Rademacher (58). Siegel first gave the proof when T(r) = -1/r, and then Rademacher gave the proof for a general modular transformation. Other gener~l methods for obtaining the trans-formation formulae for log17(r) and similar functions have been devised by T. M. Apostol (4) and B. Berndt (7), (8).

Recall that the classical theta-function 03 {r) may be defined by 00

OJ(T) = II (1- e27rinr)(1- e7ri(2n-l}r)2,

n=l

xviii BRUCE C. BERNDT

where Im T > 0. Since fh can be expressed as a quotient of three eta-functions, a transformation formula for log lh ( T) can be effected from that of the Dedekind eta-function. In one of his last papers [73], Rademacher used this idea to give a new formulation of the transformation formula for log fh ( T). A similar trans-formation formula for log (}3 ( T) involving a new analogue of Dedekind sums was discovered by Berndt [10].

The most fundamental property of Dedekind sums is the remarkable reci-procity law: If h, k > 0 and (h, k) = 1, then

(1) 1 1 (h k 1 ) s(h,k) +s(k,h) = -4 + 12 k + h + hk ·

This formula was first proved by R. Dedekind [14] using the transformation formulae of log 17( T). There now exist many proofs of ( 1), and several of these can be found in a monograph [7] on Dedekind sums, written by Rademacher and completed by Grosswald after Rademacher's death. Rademacher found five original proofs of (1). In the aforementioned paper [27], Rademacher gives an elegant proof of (1) based on the enumeration of lattice points in pyramids. An elementary proof of (1), as well as a generalization when (h, k) > 1, is given in [28]. Another elementary proof appears in a paper [44], coauthored with A. L. Whiteman. A method employing finite Fourier series was used in [32] to prove (1). In [53], Rademacher gives a particularly simple and elegant proof of (1) that depends on Riemann-Stieltjes integrals. Rademacher [57] also established the following generalization of the reciprocity law. Let a, b, and c denote pairwise coprime, positive integers. Let a', b', and c' be integers chosen so that aa' = 1 (mod be), bb' = 1 (mod ca), and cc' = 1 (mod ab). Then

(2) s(bc', a)+ s(ca', b)+ s(ab', c)=-~+ 112 (~+:a+ :b)·

Two of the most important papers in the theory of Dedekind sums are [44] and [59]. Dedekind [14] stated without proof several properties of s(h, k) in a paper that was written in order to fill some gaps in a famous paper of B. Riemann [55], first published in his "Nachlass." Rademacher and Whiteman [44] gave proofs of these properties and established some others as well, including various congruences satisfied by Dedekind sums. We mention a few of the interesting properties established by Rademacher in [59]. If 1 < h < k, then s(h, k) < s(1, k). If 0 < h < (k- 1) 112 , then s(h, k) > 0. Rademacher also showed that 12s(h, k) assumes only one integral value, namely 0, and this occurs if and only if h2 = -1 (mod k); somewhat earlier, Apostol [3] had also discovered these last two facts. Let p = h/k and put s(p) = s(h, k). Then Rademacher showed that s(p) is unbounded from above and from below on any interval in ( -oo, oo). Identities for s(h 1,k1 ) ± s(h2 ,k2 ) are found, where hdk1 and h2/k2 are consecutive fractions in a Farey sequence. Rademacher ( [8] in problems list) later asked if s(h1 , kl) + s(h2 , k2 ) is always positive. However, L. Pinzur [49]

HANS RADEMACHER xix

and K. H. Rosen [56] independently exhibited infinite classes of pairs (h1, ki), (h2, k2) for which the sum is negative.

There exists a connection between Dedekind sums and lattice points in a tetra-hedron. Let N3(a, b, c) denote the number of lattice points in the tetrahedron

(3) O:::;x<a, O:::;y<b, O:::;z<c, O<xfa+yfb+zfc<l,

where a, b, and c are relatively prime in pairs. Rademacher proved the congru-ence

(s(bc,a)- l~a) + (s(ca,b)- ;;b)+ (s(ab,c)- t:c) 1 abc 1 = -4- 12 + 12abc (mod 2),

and used a theorem of L. J. Mordell [44] to show that

(4) 1

N3(a, b, c)= 4(a + l)(b + l)(c + 1) (mod 2).

More generally consider the number of lattice points Nn(al, a2, ... , an) in then dimensional tetrahedron

0:::; Xj < aj, 1 :::; j :::; n, 0 < xda1 + x2/a2 + · · · + Xn/an < 1,

where the integers aj, 1 :::; j :::; n, are pairwise coprime. Rademacher ([6] in list of problems) conjectured that

(5)

For n = 1, 2, (5) is trivially true with the congruence sign = replaced by an equality sign. For n = 3, (5) is valid by (4). However, for n = 4, K. H. Rosen [57] found an infinite class of counterexamples to (5).

Very recently, J. E. Pommersheim [51] has removed the coprime conditions on a, b, and c and greatly extended the work of Rademacher and Mordell. Moreover, he has also found a beautiful generalization of (2).

Rademacher ([7] in problems list) posed a third problem at the number theory conference at the University of Colorado in 1963. Are the points (h/k, s(h, k)) dense in the plane? By a very elegant argument using continued fractions, D. R. Hickerson [28] answered this question affirmatively. Using results of I. Vardi [62], G. Myerson [45] proved that the sequence (hjk, s(h, k)) is uniformly distributed modulo 1, a result further refined by R. W. Bruggeman [11]. Vardi [63] more recently proved that Dedekind sums have a limiting distribution.

Generalizing a sum of F. Klein, Rademacher [70] defined the generalized Dedekind sum,

(6) s(h,k;x,y) := t. ( (h (JL;Y + ~))) ( (JL;Y)),

XX BRUCE C. BERNDT

where ( h, k) = 1 with k > 0, and where x and y are real. If x and y are integers, then (6) reduces to s(h, k). Rademacher [70] derived a beautiful reciprocity law relating s(h, k; x, y) with s(k, h; y, x).

After Rademacher and Grosswald's elegant monograph [7] on Dedekind sums was published in 1972, there appeared a profusion of papers offering many gen-eralizations. We cannot give any details here, but we remark that many elegant generalizations are due to L. Garlitz. For a survey containing some of these, see Berndt's paper [9]. Analogues or generalizations of the classical Dedekind sum s(h, k) frequently appear in transformation formulae of analogues of logry(r); perhaps the first generalization in this direction is due to Apostol [4]. Such sums also appear in formulas for values of certain zeta-functions at integral arguments.

Perhaps Rademacher's most famous theorem is his exact formula for the par-tition function p(n) [37], [38]. Let c = rrJ273 and An = Jn- 1/24. Then Rademacher proved that

(7)

where

p(n) = \'\ f Ak(n)Vkdd (sinh~An/k)), rrv2 k=l n n

Ak(n) = L Wh,ke-27rihn/k ' h (mod k) (h,k)=l

with Wh,k = exp{ rri s( h, k)}. The history of this formula is interesting. Introducing their famous circle method, Hardy and Ramanujan [24], [21, pp.

306-339], [53, pp. 276-309] had in 1917 astounded the mathematical community by finding an asymptotic series for p( n), namely,

(8)

as n tends to oo, where a is a positive constant. They started with the famil-oo

iar generating function f(x) := n (1- xn)-l for p(n), to which they applied n=l

Cauchy's integral formula on a circle centered at the origin and interior to the unit circle, and utilized a dissection of that circle by means of Farey series. Their method has two main features. First, they approximated f by a function F and integrated f - F over a dissection of the circle by a Farey series of order N. Secondly, they coupled the parameters n and N by setting N = ayln.

On the other hand, while preparing the lectures for his graduate course in analytic number theory at the University of Pennsylvania in the fall of 1936, Rademacher applied the transformation formula for the Dedekind eta-function ry(r) = e1rirll2 j j(e21rir) to obtain the principal term, and he left the parameters

HANS RADEMACHER xxi

n and N free. He subsequently found that

(9) p(n) =-1- tAk(n)Vk!£ (sinh(c>.nfk))

11'J2 k=l dn An

+ O(N-112 exp(211'nN-2 )),

as N tends to oo. Note that the principal terms are independent of N. Thus, keeping n fixed and letting N tend to oo in (9), we deduce (7).

Observe that the summands in (7) differ from those in (8). When Rademacher first obtained (7), he assumed that he had made a calculational error. On the following day, Rademacher carefully checked his analysis and found that (7) was indeed correct. D. H. Lehmer [36]later proved that the sum in (8), when extended to oo, diverges.

At about the same time Rademacher proved (7), A. Selberg also proved (7) but never published his result [60, p. 705].

The numbers Ak(n) appear in both Hardy and Ramanujan's and Rademacher's formulas, and Lehmer [37] found that they possess some remarkable factorization properties. Using their theorems on congruences for Dedekind sums, Rademacher and Whiteman [44] gave easier proofs of these factorization theorems. Selberg later discovered a much simpler formula for Ak(n), namely,

(10) L j (mod 2k)

(3j2 +j)/2=:-n (mod k)

(-1)j (11'(6j + 1)) cos 6k .

After Whiteman [67] gave the first published proof of (10), Rademacher [61] devised a simpler proof of (10) depending upon the transformation formulae for ry(r). He also gave an improved version of Lehmer's factorization theorems, while offering still easier proofs.

Another interesting theorem is proved in [37]. Let 00

f(x) = 1 + LP(n)xn, lxl < 1. n=l

Then using (7), Rademacher obtained a representation of the form 00

(11) f(x) = L fn(x), n=O

where each function fn(x) has a simple pole on lxl = 1; the closure of this set of poles, lxl = 1, constitutes the natural boundary of f(x). Furthermore, each function fn(x) has an analytic continuation to the entire complex plane. More-over, (11) represents an analytic function on lxl > 1, which, from numerical evidence, Rademacher thought might be identically zero. Rademacher ([5, pp. 190-199] in the list of Rademacher's books), in fact, later proved his conjecture.

xxii BRUCE C. BERNDT

H. Petersson [48] also proved Rademacher's conjecture, but by an entirely dif-ferent method. J. Lehner [41] established a very general theorem of which the result of Rademacher and Petersson is a special case.

The ideas in Rademacher's paper [38] were found to be even more fruitful, as he and H. S. Zuckerman [40] proved the following generalization: Let F(r) be a modular form of dimension r > 0. Suppose that F has a pole at ioo, i.e.,

00

F(r) = L n=-J.t

where Im r > 0, J-L > 0, and 0 S: a< 1. Then the coefficients an, n 2:0, are deter-mined by those with -J-L S: n S: -1. Furthermore, Rademacher and Zuckerman derived a formula for an, n 2: 0, which involves an, -J-L S: n S: -1, sums analo-gous to (10), and the modified Bessel function Ir+l· An interesting corollary is that an entire modular form of positive dimension belonging to the full modular group must be identically zero. Many other proofs of this fundamental result have been given by various writers. Zuckerman [68] found similar formulas for the coefficients of modular forms associated with congruence subgroups. In a subse-quent paper [69], Zuckerman extended the results of himself and Rademacher by establishing a corresponding theorem for modular forms of positive dimension having at most a finite number of poles in a fundamental region. Further exten-sions to more general automorphic forms were made by J. Lehner [40], [42], [43], who also generalized the aforementioned theorem on identically vanishing forms. For the special cases of the classical theta-functions, L. A. Goldberg [16] discov-ered easier proofs of Zuckerman's theorems and also derived analogues of the Selberg-Whiteman-Rademacher formula (10). Rademacher extended his ideas yet further [39] and found a formula for the Fourier coefficients of the modular invariant J(r), a modular form of dimension 0. In fact, Rademacher's formula for the Fourier coefficients of J(r) was actually derived earlier by Petersson [47] by an entirely different method. In [41], Rademacher considered the converse problem; taking the Fourier expansion of J( r) together with his formula for the coefficients, he showed that they define a modular function. The importance of this paper has been made clear by M. Knopp [31], who showed its connection with Eichler cohomology, Poincare series, and the construction of modular forms.

As previously indicated, the Hardy-Ramanujan-Littlewood circle method is a key ingredient in Rademacher's proof of his formula for p(n). Rademacher later [48] improved this method by replacing the Farey arcs by arcs of circles, called Ford circles, with centers h/k+i/(2k2 ) and radii 1/(2k2 ), where hjk runs through a Farey sequence. These new paths of integration simplify the estimates of certain integrals and clarify how the singularities contribute to the formula. A clear account of this method may be found in Apostol's book [5, pp. 99-110].

Rademacher wrote two further papers [42], [45] on p(n). In the first, coau-thored with H. S. Zuckerman, proofs are given of two famous identities of Ra-

HANS RADEMACHER xxiii

manujan,

(12) oo oo ( 1 5n)5

""'p(5n + 4)xn = 5 IJ - x ~ (1- xn)6 n=O n=l

and

(13) oo n- oo (1- x1n)3 oo (1- x1n)7 ~p(7n + 5)x - 7!! (1- xn)4 + 49x!! (1- xn)B .

In the second, Rademacher gave further proofs of these identities as well as other results of this sort. In both papers, the theory of modular forms, in particular, the transformation formulae of the Dedekind eta-function, is used. Motivated by Rademacher's ideas, M. Newman [46] established further identities akin to (12) and (13). The identities (12) and (13), respectively, immediately imply the Ramanujan congruences

(14) p(5n + 4) = 0 (mod 5)

and

(15) p(7n + 5) = 0 (mod 7).

The congruences (14) and (15), respectively, are special cases of the following Ramanujan conjectures.

(16) If 24m= 1 (mod 5n), then p(m) = 0 (mod 5n).

(17) If 24m= 1 (mod 7n), then p(m) = 0 (mod 7((n+2ll21).

The conjecture (17) is given in its corrected form; Ramanujan also conjectured a congruence for powers of 11. Although Ramanujan proved special cases of his three general conjectures, the first complete proofs of (16) and (17) were given by G. N. Watson [65] in 1938. For powers of 11, Ramanujan found proofs for n = 1, 2, and Lehner [38] used Rademacher's ideas in [45] to establish the same congruences. Lehner [39] also proved Ramanujan's conjecture for n = 3 before A. 0. L. Atkin [6], motivated by Rademacher and Lehner's ideas, first found a complete proof of the corresponding conjecture for powers of 11 in 1967. Ramanujan's unpublished manuscript on congruences for the partition function has recently been reproduced with his "lost notebook" [54]. An excellent survery of Ramanujan's work on congruences has been written by K. G. Ramanathan [52]. The books of M. Knopp [30, Chapters 7, 8] and G. E. Andrews [2, Chapter 10] contain excellent accounts of Ramanujan's congruences for p(n).

In Rademacher's last paper [74] in the area of modular forms, it is shown that T(pr) is essentially a Tschebyscheff polynomial in T(p) of the second kind, where T(n) is the Heeke operator, pis a prime, and r is a positive integer. Heeke [26], [27, pp. 644-671] had previously proved that T(pr) is a polynomial in T(p).

Before leaving modular forms, we describe Rademacher's contribution, pe-ripherally involving modular forms, to a famous problem from complex analysis.

xxiv BRUCE C. BERNDT

Let f be analytic on the unit disc lzl < 1 with f'(O) = 1. Then there exists a positive constant .C which is the least upper bound of all r such that the range off always contains a closed disc of radius r. The constant .Cis called Landau's constant. L. Ahlfors [1] proved that .C 2: 0.5, and C. Pommerenke [50] improved this slightly by showing that .C > 0.5. On the other hand, Rademacher [47] proved that .C ::=; 2nf(1/3)f(1/6)-2 = 0.54325 ... , which he conjectured to be the true value of .C. The exact value of .C is still not known.

In his papers [33], [34], and [36], Rademacher studied the geometrical distribu-tion of prime numbers w in the ring of integers of a real quadratic number field. As was the case with his papers [16], [17], and [23], the arguments involve the use of Heeke's Grossencharakteren and the corresponding Dirichlet series. For a given angular region in the (w, w') plane, where w' is the conjugate of w, Heeke [25], [27, pp. 249-289] had proved earlier that the number of prime numbers in this angular region having norm not exceeding x is asymptotic to h- 1 Li(x) times a geometric proportionality factor, where his the class number of the field and Li(x) denotes the logarithmic integral. In [33], Rademacher made Heeke's result more precise by proving it with an error term of the type obtained by de la Vallee-Poussin for the ordinary prime number theorem. Next, Rademacher [34] applied the results of [33] to obtain an analogous asymptotic formula with an error term for the number of prime numbers in a region of the form 0 < w :::; Y, 0 < w' :::; Y', where YY' > 2. In [36], he obtained the same result as in [34] by a direct argument not requiring the results of [33].

We now discuss Rademacher's papers on zeta-functions and related areas. Applying the Poisson summation formula to

00 L (a+i(n+u))-s, a>O,Re s>1, n=-oo

Rademacher [25] elegantly proved the functional equation of the Riemann zeta-function ( ( s). Using the same notion, he also established the functional equation satisfied by Dirichlet's £-functions L(s, x). These ideas have been generalized somewhat by W. Schnee [59].

As usual, put

{ logn, A(n) =

0, if n is a prime power,

otherwise.

Then, by a well-known formula (Davenport [13, p. 109]),

""'' ""' 1 2 L A(n) = x- }~moo L xP / p -log(2n)- 2log(1- x- ), n~x hi<T

where p = {3 + h runs through the non-trivial zeros of ((s), and the prime' on 1

the summation sign indicates that if x is integral, only 2A(x) is counted. The

HANS RADEMACHER XXV

infinite series on the right-hand side is closely related to

(18) L ')'-1 sin('Yy), y =log x. I

Rademacher [64] proved that if the Riemann hypothesis is true, (18) has certain jump discontinuities. He felt that by constructing a similar series with the same jumps, some light might be shed on the distribution of zeros of ((s). L.A. Rubel and E. G. Straus [58] showed that this was doubtful by constructing the type of series sought by Rademacher and proving that (18) has the aforementioned discontinuities under a much weaker assumption than the Riemann hypothesis.

In [62], [63], and [67], Rademacher proved a Phragmen-Lindelof theorem for an infinite strip that has the advantage that, when applied to problems in prime number theory, it is uniform in several parameters. He then gave applications to the growths of L(s, x) and the Dedekind zeta-function in the critical strip.

In [35], Rademacher extended the validity of a fascinating identity of F. John [29] involving a periodic function and gave applications to ((s) and Dedekind's zeta-function.

The last research paper on which we shall comment is [76]. Here, Rademacher properly interpreted and established some results stated by Euler on certain divergent series involving the pentagonal numbers n(3n-1)/2. Moreover, Rade-macher extended some of Euler's results.

Rademacher's expository papers serve not only as excellent introductions to several of his own research interests, but also to broader areas in number theory.

Analytic additive theory in algebraic number fields is examined in [54]. The problems and differences encountered when generalizing the classical theory over the rational field to algebraic number fields are discussed.

Rademacher's address [43] provides an excellent introduction to the parti-tion function p( n), including the Hardy-Littlewood-Ramanujan circle method, the asymptotic formula of Hardy and Ramanujan, and the exact formula of Rademacher. After a prefatory exposition about modular forms, Rademacher further discusses the content of his recent work [39]-[42].

Rademacher did not deem function theory to be merely a tool in studying properties of numbers. "It is more the inner harmony of a system" with both disciplines bearing on the character of the other. This belief iE fully developed by Rademacher in [46]. "The name 'analytic number theory' implies, as I take it, a thorough fusion of analysis and arithmetic, in which as we shall see, analy-sis is not necessarily subordinate to arithmetic." Often analytic number theory is regarded as the development of asymptotic formulas. However, Rademacher believed that identities, group-theoretical arguments, and structural considera-tions have just as important roles in the subject. Rademacher provides a brief, eloquent history of analytic number theory in [46], wherein the names of Dirich-let, Riemann, Landau, Hardy, Littlewood, Ramanujan, and many others are highlighted. We must now add to this roll of honor the name of Rademacher.

xxvi BRUCE C. BERNDT

The author is very grateful to Paul Bateman, Heini Halberstam, Marvin Knopp, Joseph Lehner, and Andrzej Schinzel for many helpful contributions.

This paper is a slightly modified version of an article that first appeared in Acta Arithmetica 61 (1992), 209-231. We are grateful to the editors of Acta Arithmetica for permission to republish the paper in this memorial volume to Hans Rademacher.

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4. T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147-157.

5. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976.

6. A. 0. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), pp. 67-78.

7. B. C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums, Trans. Amer. Math. Soc. 178 (1973), pp. 495-508.

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9. B. C. Berndt, Reciprocity theorems for Dedekind sums and generalizations, Advances Math. 23 (1977), pp. 285-316.

10. B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. Reine Angew. Math. 304 (1978), pp. 332-365.

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12. Y. Chuman, Generators and relations of f 0 (N), J. Math. Kyoto Univ. 13 (1973), pp. 381-390.

13. H. Davenport, Multiplicative Number Theory, second ed., Springer-Verlag, New York, 1980.

14. R. Dedekind, Erlauterungen zu zwei Fragmenten von Riemann, Gesammelte Mathematische Werke, Friedr. Vieweg & Sohn, Braunschweig, 1930, pp. 159-172.

HANS RADEMACHER xxvii

15. H. Frasch, Die Erzeugenden der Hauptkongruenzgruppen fUr Primzahlstufen, Math. Ann. 108 (1933), 229-252.

16. L. A. Goldberg, Transformations of Theta-functions and Analogues of Dede-kind Sums, Ph.D. Dissertation, University of Illinois at Urbana-Champaign, 1981.

17. E. Grosswald, On the structure of some subgroups of the modular group, Amer. J. Math. 72 (1950), 809-834.

18. E. Grosswald, On the parabolic generators of the principal congruence sub-groups of the modular group, Amer. J. Math. 74 (1952), 435-443.

19. E. Grosswald, An orthonormal system and its Lebesgue constants, Analytic Number Theory, M. I. Knopp, editor, Lecture Notes in Math. No. 899, Springer-Verlag, Berlin, 1981, pp. 2-9.

20. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, New York, 1974.

21. G. H. Hardy, Collected Papers, vol. 1, Clarendon Press, Oxford, 1966.

22. G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approxi-mation. I. The fractional part of nk(), Acta Math. 37 (1914), pp. 155-191.

23. G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio Numerorum': III. On the expression of a number as a sum of primes, Acta. Math. 44(1922), 1-70.

24. G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analy-sis, Proc. London Math. Soc. (2) 17 (1918), pp. 75-115.

25. E. Heeke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteil-ung der Primzahlen. Zweite Mitteilung, Math. Z. 6(1920), 11-51.

26. E. Heeke, Uber Modulfunktionen und die Dirichletschen Reihen mit Euler-scher Produktenwicklung. I, Math. Ann 114 (1937), pp. 1-28.

27. E. Heeke, Mathematische Werke, Vandenhoeck & Ruprecht, Gottingen, 1970.

28. D. R. Hickerson, Continued fractions and density results for Dedekind sums, J. Reine Angew. Math. 290 (1977), pp. 113-116.

29. F. John, Identitiiten zwischen dem Integral einer willkiirlichen Funktion und unendlichen Reihen, Math. Ann. 110 (1935), pp. 718-721.

30. M. Knopp, Modular Functions in Analytic Number Theory, Markham, Chicago, 1970.

31. M. Knopp, Rademacher on J(T), Poincare series of nonpositive weights and the Eichler cohomology, Notices Amer. Math. Soc. 37(1990), 385-393.

xxviii BRUCE C. BERNDT

32. 0. Korner, Ubertragung des Goldbach-Vinogradovschen Satzes auf reell-quad-ratische Zahlkorper, Math. Ann. 141(1960), 343-366.

33. 0. Korner, Erweiterter Goldbach-Vinogradovscher Satz in beliebigen alge-braischen Zahlkorpern, Math. Ann. 143(1961), 344-378.

34. 0. Korner, Zur additiven Primzahltheorie algebraischer Zahlkorper, Math. Ann. 144(1961), 97-109.

35. R. S. Kulkarni, An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113(1991), 1053-1133.

36. D. H. Lehmer, The Hardy-Ramanujan series for the partition function, J. London Math. Soc. 12(1937), 171-176.

37. D. H. Lehmer, On the series for the partition function, Trans. Amer. Math. Soc. 43 (1938), pp. 271-295.

38. J. Lehner, Ramanujan identities involving the partition function for the mod-uli 11 a:, Amer. J. Math. 65(1943), 492-520.

39. J. Lehner, Proof of Ramanujan's partition congruence for the modulus 113 ,

Proc. Amer. Math. Soc. 1(1950), 172-181.

40. J. Lehner, The Fourier coefficients of automorphic forms belonging to a class of horocyclic groups, Michigan Math. J. 4 (1957), 265-279.

41. J. Lehner, Partial fraction decompositions and expansions of 0, Trans. Amer. Math. Soc. 87 (1958), 130-143.

42. J. Lehner, The Fourier coefficients of automorphic forms on horocyclic groups, II, Michigan Math. J. 6 (1959), 173-193.

43. J. Lehner, The Fourier coefficients of automorphic forms on horocyclic groups, III, Michigan Math. J. 7 (1960), 65-74.

44. L. J. Mordell, Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian Math. Soc. 15 (1951), pp. 41-46.

45. G. Myerson, Dedekind sums and uniform distribution, J. Number Theory 28 (1988), pp. 233-239.

46. M. Newman, Remarks on some modular identities, Trans. Amer. Math. Soc. 73(1952), 313-320.

47. H. Petersson, Uber die Entwicklungskoeffizienten der automorphen Formen, Acta Math. 58 (1932), pp. 169-215.

48. H. Petersson, Die linearen Relationen zwischen den ganzen Poincareschen Reihen von reeller Dimension zur Modulgruppe, Abh. Math. Sem. Univ. Hamburg 12 (1938), 415-472.

HANS RADEMACHER xxix

49. L. Pinzur, On a question of Rademacher concerning Dedekind sums, Proc. Amer. Math. Soc. 61 (1976), pp. 11-15.

50. C. Pommerenke, On Bloch functions, J. London Math. Soc. (2) 2 (1970), pp. 689-695.

51. J. E. Pommersheim, Toric varieties, lattice points, and Dedekind sums, Math. Ann. 295 (1993), 1-24.

52. K. G. Ramanathan, Ramanujan and the congruence properties of partitions, Proc. Indian Acad. Sci. (Math. Sci.) 89 (1980), pp. 133-157.

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54. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

55. B. Riemann, Fragmente iiber die Grenzfalle der elliptischen Modulfunktionen, Gesammelte Mathematische Werke, Dover, New York, 1953, pp. 455-465.

56. K. H. Rosen, On the sign of some Dedekind sums, J. Number Theory 9 (1977), pp. 209-212.

57. K. H. Rosen, Lattice points in four-dimensional tetrahedra and a conjecture of Rademacher, J. Reine Angew. Math. 307/308 (1979), pp. 264-275.

58. L. A. Rubel and E. G. Straus, Special trigonometric series and the Riemann hypothesis, Math. Scand. 18 (1966), pp. 35-44.

59. W. Schnee, Die Funktionalgleichung der Zetafunktion und der Dirichletschen Reihen mit periodischen Koeffizienten, Math. Z. 31 (1930), pp. 378-390.

60. A. Selberg, Reflections around the Ramanujan centenary, Collected Papers, vol. 1, Springer-Verlag, Berlin, 1989, pp. 695-706.

61. C. L. Siegel, A simple proof of 17( -1/r) = TJ(r)JT!i, Mathematika 1 (1954), p. 4.

62. I. Vardi, A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987), 189-197.

63. I. Vardi, Dedekind sums have a limiting distribution, Inter. Math. Res. Not., Duke Math. J. 69 (1993), 1-12.

64. J. L. Walsh, A closed set of normal, orthogonal functions, Amer. J. Math. 55 (1923), pp. 5-24.

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XXX BRUCE C. BERNDT

67. A. Whiteman, A sum connected with the series for the partition function, Pacific J. Math. 6 (1956), pp. 159-176.

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Papers by Hans Rademacher

1. Eineindeutige Abbildungen und Messbarkeit, Monats. Math. Phys. 27 (1916), pp. 183-291.

2. (with C. Caratheodory) Uber die Eineindeutigkeit im Kleinen und im Crossen stetiger Abbildungen von Gebieten, Arch. Math. Phys. (3) 26 (1917), pp. 1-9.

3. Zu dem Borelschen Satz iiber die asymptotische Verteilung der Ziffern in Dezimalbriichen, Math. Z. 2 (1918), pp. 306-311.

3a. Nachtriigliche Bemerkung zu meiner Arbeit "Uber die asymptotische Verteil-ung der Ziffern in Dezimalbriichen," Math. Z. 3 (1919), p. 317.

4. Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Vari-abeln und iiber die Transformation der Doppelintegrale, Math. Ann. 79 (1919), pp. 340-359.

5. Uber streckentreue und winkeltreue Abbildung, Math. z. 4 (1919), pp. 131-138.

6. Bemerkungen zu den Cauchy-Riemannschen Differentialgleichungen und zum Moreraschen Satz, Math. Z. 4 (1919), pp. 177-185.

7. Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Vari-abeln. II, Math. Ann. 81 (1920), pp. 52-63.

8. Zur Theorie der Minkowskischen Stiitzebenenfunktio~, S.-B. Berlin Math. Gesell. 20 (1921), pp. 14-19.

9. Uber eine Eigenschaft von messbaren Mengen positiven Masses, Jber. Deutsch. Math.-Verein. 30 (1921), pp. 130-132.

10. Uber die asymptotische Verteilung gewisser konvergenzerzeugender Faktoren, Math. z. 11 (1921), pp. 276-288.

11. Uber den Konvergenzbereich der Eulerschen Reihentransformation, S.-B. Ber-lin Math. Gesell. 21 (1922), pp. 16-24.

12. Uber eine funktionale Ungleichung in der Theorie der konvexen Korper, Math. z. 13 (1922), pp. J.8-27.

HANS RADEMACHER xxxi

13. Einige Satze iiber Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), pp. 112-138.

14. Beitrage zur Viggo Brunschen Methode in der Zahlentheorie, Abh. Math. Sem. Univ. Hamburg 3 (1923), pp. 12-30.

15. Uber die Anwendung der Viggo Brunschen Methode auf die Theorie der alge-braischen Zahlkorper, S.-B. Preus. Akad. Wiss. Phys.-Math. Kl. 24 (1923), pp. 211-218.

16. Zur additiven Primzahltheorie algebraischer Zahlkorper. I. Uber die Darstel-lung totalpositiver Zahlen als Summe von totalpositiven Primzahlen im reell-quadratischen Zahlkorper, Abh. Math. Sem. Univ. Hamburg 3 (1924), pp. 109-163.

17. Zur additiven Primzahltheorie algebraischer Zahlkorper. II. Uber die Darstel-lung von Korperzahlen als Summe von Primzahlen im imaginar-quadratischen Zahlkorper, Abh. Math. Sem. Univ. Hamburg 3 (1924), pp. 331-378.

18. Uber den Vektorenbereich eines konvexen ebenen Bereiches, Jber. Deutsch. Math.-Verein. 34 (1925), pp. 64-79.

19. Uber eine Erweiterung des Goldbachschen Problems, Math. Z. 25 (1926), pp. 627-657.

20. Eine Bemerkung zu Herrn Peterssons Arbeit: Uber die Darstellung natiirlicher Zahlen durch definite und indefinite quadratische Formen von 2r Variablen," Abh. Math. Sem. Univ. Hamburg 5 (1926), pp. 40-44.

21. (with F. Reiche) Die Quantelung des symmetrischen Kreisels nach Schrodingers Undulationsmechanik, z. Phys. 39 (1926), pp. 444-464.

22. (with F. Reiche) Die Quantelung des symmetrischen Kreisels nach Schrodingers Undulationsmechanik. II. lntensitatsfragen, Z. Phys. 41 (1927), pp. 453-492.

23. Zur additiven Primzahltheorie algebraischer Zahlkorper. III. Uber die Darstel-lung totalpositiver Zahlen als Summen von totalpositiven Primzahlen in einem beliebigen Zahlkorper, Math. Z. 27 (1928), pp. 321-426.

24. Zur Theorie der Modulfunktionen, Atti del Congresso Internazionale dei Mat-ematici, Bologna, 1928, vol. 3, Nicola Zanichelli, Bologna, 1928, pp. 297-301.

25. Ein neuer Beweis fiir die Funktionalgleichung der ( -Funktion, Math. Z. 31 (1929), pp. 39-44.

26. Uber die Erzeugenden von Kongruenzuntergruppen der Modulgruppe, Abh. Math. Sem. Univ. Hamburg 7 (1929), pp. 134-148.

27. Zur Theorie der Modulfunktionen, J. Reine Angew. Math. 167 (1932), pp. 312-336.

xxxii BRUCE C. BERNDT

28. Eine arithmetische Summenformel, Monats. Math. Phys. 39 (1932), pp. 221-228.

29. Bestimmung einer gewissen Einheitswurzel in der Theorie der Modulfunktio-nen, J. London Math. Soc. 7 (1932), pp. 14-19.

30. Lebensbild von Wilhelm Foerster, Schlesische Lebensbilder 3 (1932), pp. 343-348.

31. Mathematische Theorie der Genkoppelung unter Beriicksichtigung der Inter-ferenz, Schlesische Gesell. Vaterlandische Cultur (1932), pp. 1-8.

32. Egy Reciprocitaskepletrol a Modulfiiggevenyek ElmeletebOl, Mat. Fiz. Lapok 40 (1933), pp. 24-34.

33. Primzahlen reell-quadratischer Zahlkorper in Winkelriiumen, Math. Ann. 111 (1935), pp. 209-228.

34. Uber die Anzahl der Primzahlen eines reell-quadratischen Zahlkorpers, deren Konjugierte unterhalb gegebener Grenzen liegen, Acta Arith. 1 (1935), pp. 67-77.

35. Some remarks on F. John's identity, Amer. J. Math. 58 (1936), pp. 169-176.

36. On prime numbers of real quadratic fields in rectangles, 'frans. Amer. Math. Soc. 39 (1936), pp. 380-398.

37. A convergent series for the partition function p(n), Proc. Nat. Acad. Sci. U.S.A. 23 (1937), pp. 78-84.

38. On the partition function p(n), Proc. London Math. Soc. (2) 43 (1937), pp. 241-254.

39. The Fourier coefficients of the modular invariant J(r), Amer. J. Math. 60 (1938), pp. 501-512.

40. (with H. S. Zuckerman) On the Fourier coefficients of certain modular forms of positive dimension, Ann. of Math. 39 (1938), pp. 433-462.

41. The Fourier series and the functional equation of the absolute modular inve-riant J(r), Amer. J. Math. 61 (1939), pp. 237-248.

41a. Correction, Amer. J. Math. 64 (1942), p. 456.

42. (with H. S. Zuckerman) A new proof of two of Ramanujan's identities, Ann. of Math. 40 (1939), pp. 473-489.

43. Fourier expansions of modular forms and problems of partition, Bull. Amer. Math. Soc. 46 (1940), pp. 59-73.

44. (with A. Whiteman) Theorems on Dedekind sums, Amer. J. Math. 63 (1941), pp. 377-407.

HANS RADEMACHER xxxiii

45. The Ramanujan identities under modular substitutions, Trans. Amer. Math. Soc. 51 (1942), pp. 609-636.

46. Trends in research: the analytic number theory, Bull. Amer. Math. Soc. 48 (1942), pp. 379-401.

47. On the Bloch-Landau constant, Amer. J. Math. 65 (1943), pp. 387-390.

48. On the expansion of the partition function in a series, Ann. of Math. 44 (2) (1943), pp. 416-422.

49. (with I. J. Schoenberg) An iteration method for calculation with Laurent series, Quart. Appl. Math. 4 (1946), pp. 142-159.

50. On the accumulation of errors in processes of integration on high-speed calcu-lating machines, Proceedings of a Symposium on Large-scale Digital Calculat-ing Machinery, Annals of the Computation Laboratory of Harvard University, vol. 16, Harvard Univ. Press, Cambridge, Mass., 1948, pp. 176-187.

51. On a theorem of Frobenius, Studies and Essays Presented toR. Courant on His 60th Birthday, January 8, 1948, Interscience, New York, 1948, pp. 301-305.

52. (with I. J. Schoenberg) Helly's theorems on convex domains and Tchebycheff's approximation problem, Canad. J. Math. 2 (1950), pp. 245-256.

53. Die Reziprozitiitsformel fiir Dedekindsche Summen, Acta Sci. Math. (Szeged) 12, Part B (1950), pp. 57-60. '

54. Additive algebraic number theory, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, 1952, pp. 356-362.

55. On the condition of Riemann integrability, Amer. Math. Monthly 61 (1954), pp. 1-8.

56. On Dedekind sums and lattice points in a tetrahedron, Studies in Mathe-mataics and Mechanics Presented to Richard von Mises, Academic Press, New York, 1954, pp. 49-53.

57. Generalization of the reciprocity formula for Dedekind sums, Duke Math. J. 21 (1954), pp. 391-397.

58. On the transformation of log 'IJ(T), J. Indian Math. Soc. 19 (1955), pp. 25-30.

59. Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956), pp. 445-463.

60. Fourier analysis in number theory, Final Technical Report, Symposium on Harmonic Analysis and Related Integral Transforms, Cornell Univ., Ithaca, New York, 1956, 25 pages.

61. On the Selberg formula for Ak(n), J. Indian Math. Soc. 21 (1957), pp. 41-55.

xxxiv BRUCE C. BERNDT

62. On the Phragmen-Lindelof theorem and subharmonic functions, Technical Report, Cornell Univ., Ithaca, New York, 1957, 10 pages.

63. On the Phragmen-Lindelof theorem and some applications, Seminars on An-alytic Functions, vol. 1, Institute for Advanced Study, Princeton, 1958, pp. 314-322.

64. Remarks concerning the Riemann-von Mangoldt formula, Report of the Insti-tute in the Theory of Numbers, Univ. of Colorado, Boulder, Colorado, 1959, pp. 31-37.

65. A fundamental theorem in the theory of modular functions, Report of the Institute in the Theory of Numbers, Univ. of Colorado, Boulder, Colorado, 1959, pp. 71-72.

66. On the Hurwitz zetafunction, Report of the Institute in the Theory of Num-bers, Univ. of Colorado, Boulder, Colorado, 1959, pp. 73-77.

67. On the Phragmen-Lindelof theorem and some applications, Math. Z. 72 (1959/60), pp. 192-204.

68. A proof of a theorem on modular functions, Amer. J. Math. 82 (1960), pp. 338-340.

69. On a theorem of Besicovitch, Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, California, 1962, pp. 294-296.

70. Some remarks on certain generalized Dedekind sums, Acta Arith. 9 (1964), pp. 97-105.

71. On the number of certain types of polyhedra, Illinois J. Math. 9 (1965), pp. 361-380.

72. (with R. Dickson and M. Plotkin) A packing problem for parallelepipeds, J. Comb. Theory 1 (1966), pp. 3-14.

73. Uber die Transformation der Logarithmen der Thetafunktionen, Math. Ann. 168 (1967), pp. 142-148.

74. Eine Bemerkung iiber die Heckeschen Operatoren T(n), Abh. Math. Sem. Univ. Hamburg 31 (1967), pp. 149-151.

75. On the roundest oval, Trans. New York Acad. Sci. (2) 29 (1967), pp. 868-874.

76. Comments on Euler's "De mirabilibus proprietatibus numerorum pentagonal-ium," Abhandlungen aus Zahlentheorie und Analysis, ed. by P. 1\min, VEB Deutscher Verlag Wiss., Berlin, 1968, pp. 257-268.

HANS RADEMACHER XXXV

Books and Lecture Notes by Hans Rademacher

1. (with 0. Toeplitz) Von Zahlen und Figuren, J. Springer, Berlin, 1930; English edition, The Enjoyment of Mathematics, trans. by H. S. Zuckerman (who added two chapters), Princeton University Press, Princeton, 1957.

2. (with E. Steinitz) Vorlesungen iiber die Theorie der Polyeder unter Einschluss der Elemente der Topologie, Grundlehren Math. Wiss., Band 41, J. Springer, Berlin, 1934.

3. Lecture Notes on Elementary Mathematics from an Advanced Viewpoint, Eugene, Oregon, 1954.

4. Lecture Notes on Analytic Additive Number Theory, Eugene, Oregon, 1954.

5. Lectures on Analytic Number Theory, Tata Inst. of Fundamental Research, Bombay, 1954/55.

6. Lectures on Elementary Number Theory, Blaisdell, New York, 1964.

7. (with E. Grosswald) Dedekind Sums, Carus Math. Monograph No. 16, Math. Assoc. of Amer., Washington, D. C., 1972.

8. Topics in Analytic Number Theory, Grundlehren Math. Wiss., Band 169, Springer-Verlag, New York, 1973.

9. Collected Papers of Hans Rademacher (2 volumes), The MIT Press, Cam-bridge, Mass., 1974.

10. Higher Mathematics from an Elementary Point of View, ed. by D. Goldfeld, Birkhauser, Boston, 1982.

Problems and Solutions to Problems by Hans Rademacher

1. Aufgabe 26, Jber. Deutsch. Math.-Verein. 34 (1925-26), p. 98; solutions by M. Krafft and G. Szego, Jber. Deutsch. Math.-Verein. 35 (1926), pp. 50-52.

2. Aufgabe 30, Jber. Deutsch. Math.-Verein. 34 (1925-26), p. 158; solution by A. Brauer, Jber. Deutsch. Math.-Verein. 35 (1926), pp. 92-94.

3. Aufgabe 31, Jber. Deutsch. Math.-Verein. 34 (1925-26), p. 158; solution by A. Brauer, Jber. Deutsch. Math.-Verein. 35 (1926), pp. 94-95.

4. Aufgabe 32, Jber. Deutsch. Math.-Verein. 34 (1925-26), pp. 158-159; solu-tion by A. Brauer, Jber. Deutsch. Math.-Verein. 35 (1926), pp. 95-96.

5. Solution to Aufgabe 184 posed by T. Nagell, Jber. Deutsch. Math.-Verein. 46 (1936), pp. 4-5.

6. Problem 98. Number of lattice points in a tetrahedron, Proceedings of the 1963· Number Theory Conference, Univ. of Colorado, Boulder, Colorado, 1963, pp. 111-112.

xxxvi BRUCE C. BERNDT

7. Problem 99. Density of the points (h/k, s(h, k)), Proceedings of the 1963 Number Theory Conference, Univ. of Colorado, Boulder, Colorado, 1963, p. 112.

8. Problem 100. Positivity of Dedekind sums, Proceedings of the 1963 Number Theory Conference, Univ. of Colorado, Boulder, Colorado, 1963, p. 112.

Department of Mathematics University of Illinois 1409 West Green Street Urbana, Illinois 61801 U.S.A.

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applications in China, 1991 117 Morton Brown, Editor, Continuum theory and dynamical systems, 1991 116 Brian Harboorne and Robert Speiser, Editors, Algebraic geometry: Sundance 1988, 1991 115 Nancy Flournoy and Robert K. Tsutakawa, Editors, Statistical multiple integration, 1991 114 Jeffrey C. Lagarias and Michael J. Todd, Editors, Mathematical developments arising

from linear programming, 1990 113 Eric Grinberg and Eric Todd Quinto, Editors, Integral geometry and tomography, 1990 112 Philip J. Brown and Wayne A. Fuller, Editors, Statistical analysis of measurement error

models and applications, 1990 111 Earl S. Kramer and Spyros S. Magliveras, Editors, Finite geometries and combinatoricJ

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theory, 1990

(See the AMS catalog for earlier titles)

The Rademacher Legacy to Mathematics George E. Andrews, David M. Bressoud,

and L. Alayne Parson, Editors This book contains papers presented at the Hans Rademacher Centenary

Conference, held at Pennsylvania State University in July 1992. The astonish-ing breadth of Rademacher's mathematical interests is well represented in this volume. The papers collected here range over such topics as modular forms, par-titions and q-series, Dedekind sums, and Ramanujan type identities. Rounding out the volume is the opening paper, which presents a biography of Rademacher. This volume is a fitting tribute to a remarkable mathematician whose work con-tinues to influence mathematics today.

ISBN 0-8218-5173-X

9 780821 851739