Number Theory in the Spirit of Ramanujan pdf epub mobi txt 電子書 下載2025

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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