Representations of Integers as Sums of Squares

Representations of Integers as Sums of Squares

Emil Grosswald Representations of Integers as Sums of Squares Springer-Verlag New York Berlin Heidelberg Tokyo

Emil Grosswald Temple University College of Liberal Arts Philadelphia, PA 19122 U.S.A. With 6 Illustrations AMS Classifications: 10-01, WB05, WB35, loc05, 10C15 Library of Congress Cataloging in Publication Data Grosswald, Emil. Representations of integers as sums of squares. Bibliography: p. Includes index. 1. Numbers, Natural. 2. Sequences (Mathematics) 3. Forms, Quadratic. I. Title. QA246.5.G76 1985 512'.7 85-4664 1985 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1985 All rights reserved. No part ofthis book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. Typeset by Asco Trade Typesetting Ltd., Hong Kong. 9 8 7 6 543 2 1 ISBN-13: 978-1-4613-8568-4 DOl: 10.1007/978-1-4613-8566-0 e-isbn-13: 978-1-4613-8566-0

Preface During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but consisted of particularly gifted and eager listeners; unfortunately, their background varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the intention of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be. Specifically, I had read with much pleasure a book by K. Zeller on Summability (Ergebnisse der Mathematik und ihrer Grenzgebiete No. 15, Springer Verlag). What impressed me mainly was the completeness of the bibliographic references. I was moved to emulate this model and write a book on representations by sums of squares that would quote a comfortably large number of known results, occasionally with condensed proofs only, but with bibliographic references as complete as possible. Professor Rota encouraged me to write such a text, and I proceeded. When the manuscript was completed, however, it came as a real surprise to me that, except for the attempt to have a complete bibliography, there was no resemblance whatsoever between my text and its model by Zeller. The original draft profited greatly from suggestions made by Professors George Andrews (The Pennsylvania State University), Marvin Knopp (Temple University), and Olga Taussky-Todd (California Institute of Technology), as well as by an anonymous referee. Also, Professor Martin Kneser (University of Gottingen) read the whole manuscript at least twice, with incredible care, pointing out a large number of errors of omission as well as of commission. To

vi Preface all of them I express my deepest gratitude. Particular thanks are due to all colleagues, who called my attention to bibliographic items which had eluded me. I also thank Professor Rota; his encouragement was an essential element in the decision to develop my notes into the present text. At a certain moment the original publisher appeared to have lost interest in this venture. I am happy that Springer-Verlag was receptive to the suggestion that it take over. Perhaps it is appropriate that the publishers of Limitierungsverfahren... and of "Representations of integers as sums of squares" should be the same. I express my gratitude to Springer-Verlag for its support and cooperation. Finally, I remember fondly my audience at the Technion: their keen interest was an important stimulus in the preparation of the notes that grew into this manuscript. My visit at the Technion had been made possible by a Lady Davis Fellowship, for which I also express my gratitude. May the reader have as much fun from this volume as the author had in writing it! Narberth, Pennsylvania May 22,1984 EMIL GROSSW ALD

Contents Preface v Introduction CHAPTER 1 Preliminaries 1. The Problems of Representations and Their Solutions 2. Methods 3. The Contents of This Book 4. References 5. Problems 6. Notation CHAPTER 2 Sums of Two Squares 1. The One Square Problem 2. The Two Squares Problem 3. Some Early Work 4. The Main Theorems 5. Proof of Theorem 2 6. Proof of Theorem 3 7. The "Circle Problem" 8. The Determination of N 2 (x) 9. Other Contributions to the Sum of Two Squares Problem 1O. Problems 5 5 6 9 11 11 11 13 13 13 14 15 16 18 20 22 22 23 CHAPTER 3 Triangular Numbers and the Representation of Integers as Sums of Four Squares 24 1. Sums of Three Squares 24 2. Three Squares, Four Squares, and Triangular Numbers 25 3. The Proof of Theorem 2 27 4. Main Result 30 5. Other Contributions 31 6. Proof of Theorem 4 31 7. Proof of Lemma 3 33

Vlll 8. Sketch of Jacobi's Proof of Theorem 4 9. Problems CHAPTER 4 Representations as Sums of Three Squares 1. The First Theorem 2. Proof of Theorem 1, Part I 3. Early Results 4. Quadratic Forms 5. Some Needed Lemmas 6. Proof of Theorem 1, Part II 7. Examples 8. Gauss's Theorem 9. From Gauss to the Twentieth Century 10. The Main Theorem 11. Some Results from Number Theory 12. The Equivalence of Theorem 4 with Earlier Formulations 13. A Sketch of the Proof of (4.7') 14. Liouville's Method 15. The Average Order of r3(n) and the Number of Representable Integers 16. Problems CHAPTER 5 Legendre's Theorem 1. The Main Theorem and Early Results 2. Some Remarks and a Proof That the Conditions Are Necessary 3. The Hasse Principle 4. Proof of Sufficiency of the Conditions of Theorem 1 5. Problems Contents 35 36 38 38 38 39 39 42 46 49 51 53 54 55 57 59 60 61 64 66 66 67 68 68 71 CHAPTER 6 Representations ofintegers as Sums of Nonvanishing Squares 72 1. Representations by k ;:;. 4 Squares 72 2. Representations by k Nonvanishing Squares 72 3. Representations as Sums of Four Nonvanishing Squares 74 4. Representations as Sums of Two Nonvanishing Squares 75 5. Representations as Sums of Three Nonvanishing Squares 75 6. On the Number ofintegers n ~ x That Are Sums of k Nonvanishing Squares 79 7. Problems 83 CHAPTER 7 The Problem of the Uniqueness of Essentially Distinct Representations 1. The Problem 2. Some Preliminary Remarks 84 84 85

Contents 3. The Case k = 4 4. The Case k ;;:, 5 5. The Cases k = 1 and k = 2 6. The Case k = 3 7. Problems CHAPTER 8 Theta Functions 1. Introduction 2. Preliminaries 3. Poisson Summation and Lipschitz's Formula 4. The Theta Functions 5. The Zeros of the Theta Functions 6. Product Formulae 7. Some Elliptic Functions 8. Addition Formulae 9. Problems CHAPTER 9 Representations of Integers as Sums of an Even Number of Squares 1. A Sketch of the Method 2. Lambert Series 3. The Computation of the Powers O~k 4. Representation of Powers of 0 3 by Lambert Series 5. Expansions of Lambert Series into Divisor Functions 6. The Values of the rk(n) for Even k ",:; 12 7. The Size of rdn) for Even k ",:; 8 8. An Auxilliary Lemma 9. Estimate of rlo(n) and r12(n) 1O. An Alternative Approach 11. Problems CHAPTER 10 Various Results on Representations as Sums of Squares 1. Some Special, Older Results 2. More Recent Contributions 3. The Multiplicativity Problem 4. Problems CHAPTER 11 Preliminaries to the Circle Method and the Method of Modular Functions 1. Introduction 2. Farey Series 3. Gaussian Sums 4. The Modular Group and Its Subgroups IX 85 86 87 88 89 91 91 91 92 95 97 99 101 104 105 107 107 108 112 114 117 121 121 123 124 126 127 128 128 129 131 133 134 134 136 137 139

x 5. Modular Forms 6. Some Theorems 7. The Theta Functions as Modular Forms 8. Problems Contents 143 145 146 147 CHAPTER 12 The Circle Method 1. The Principle of the Method 2. The Evaluation of the Error Terms and Formula for r5(n) 3. Evaluation of the Singular Series 4. Explicit Evaluation of 9" 5. Discussion of the Density of Representations 6. Other Approaches 7. Problems 149 149 153 156 160 170 173 173 CHAPTER 13 Alternative Methods for Evaluating r,(n) 175 l. Estermann's Proof 175 2. Sketch of the Proof by Modular Functions 178 3. The Function 1/15(7:) 180 4. The Expansion of 1/15(7:) at the Cusp 7: = -1 182 5. The Function e5(7:) 184 6. Proof of Theorem 4 184 7. Modular Functions and the Number of Representations by Quadratic Forms 185 8. Problems 186 CHAPTER 14 Recent Work 188 l. Introduction 188 2. Notation and Definitions 189 3. The Representation of Totally Positive Algebraic Integers as Sums of Squares 192 4. Some Special Results 194 5. The Circle Problem in Algebraic Number Fields 196 6. Hilbert's 17th Problem 197 7. The Work of Artin 198 8. From Artin to Pfister 201 9. The Work of Pfister and Related Work 203 10. Some Comments and Additions 207 11. Hilbert's 11th Problem 208 12. The Classification Problem and Related Topics 209 13. Quadratic Forms Over Q p 212 14. The Hasse Principle 216

Contents Xl APPENDIX Open Problems 219 References 221 Bibliography 231 Addenda 242 Author Index 243 Subject Index 247