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Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of q

-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics.

The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

Bruce C. Berndt: University of Illinois, Urbana-Champaign, Urbana, IL

Cover 1
Title 4
Copyright 5
Contents 6
Preface 10
Chapter 1. Introduction 22
§1.1. Notation and Arithmetical Functions 22
§1.2. What are q- Series and Theta Functions? 27
§1.3. Fundamental Theorems about q-Series and Theta Functions 28
§1.4. Notes 43
Chapter 2. Congruences for p(n) and r(n) 48
§2.1. Historical Background 48
§2.2. Elementary Congruences for r(n) 49
§2.3. Ramanujan's Congruence p(5n + 4) = 0 (mod 5) 52
§2.4. Ramanujan's Congruence p(7n + 5) = 0 (mod 7) 60
§2.5. The Parity of p(n) 64
§2.6. Notes 70
Chapter 3. Sums of Squares and Sums of Triangular Numbers 76
§3.1. Lambert Series 76
§3.2. Sums of Two Squares 77
§3.3. Sums of Four Squares 80
§3.4. Sums of Six Squares 84
§3.5. Sums of Eight Squares 88
§3.6. Sums of Triangular Numbers 92
§3.7. Representations of Integers by x[(sup)2] + 2y[(sup)2], x[(sup)2] + 3y[(sup)2], and x[(sup)2] + xy + y[(sup)2] 93
§3.8. Notes 100
Chapter 4. Eisenstein Series 106
§4.1. Bernoulli Numbers and Eisenstein Series 106
§4.2. Trigonometric Series 108
§4.3. A Class of Series from Ramanujan's Lost Notebook Expressible in Terms of P, Q, and R 118
§4.4. Proofs of the Congruences p(5n + 4) = 0 (mod 5) and p(7n + 5) = 0(mod7) 123
§4.5. Notes 126
Chapter 5. The Connection Between Hypergeometric Functions and Theta Functions 130
§5.1. Definitions of Hypergeometric Series and Elliptic Integrals 130
§5.2. The Main Theorem 135
§5.3. Principles of Duplication and Dimidiation 141
§5.4. A Catalogue of Formulas for Theta Functions and Eisenstein Series 143
§5.5. Notes 149
Chapter 6. Applications of the Primary Theorem of Chapter 5 154
§6.1. Introduction 154
§6.2. Sums of Squares and Triangular Numbers 155
§6.3. Modular Equations 161
§6.4. Notes 171
Chapter 7. The Rogers–Ramanujan Continued Fraction 174
§7.1. Definition and Historical Background 174
§7.2. The Convergence, Divergence, and Values of R(q) 176
§7.3. The Rogers–Ramanuj an Functions 179
§7.4. Identities for R(q) 182
§7.5. Modular Equations for R(q) 187
§7.6. Notes 188
Bibliography 192
Index 206
Back Cover 210
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