Number Theory in the Spirit of Ramanujan (Student Mathematical Library)

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. Front Cover 1 Back Cover 2 Contents 3 Preface 7 Chapter 1 – Introduction 19 1.1. Notation and Arithmetical Functions 19 1.2. What are q-Series and Theta Functions? 24 1.3. Fundamental Theorems about q-Series and Theta Functions 25 1.4. Notes 40 Chapter 2 – Congruences for p(n) and τ(n) 45 2.1. Historical Background 45 2.2. Elementary Congruences for τ(n) 46 2.3. Ramanujan's Congruence p(5n + 4) o 0 (mod 5) 49 2.4. Ramanujan's Congruence p(7n + 5) o 0 (mod 7) 57 2.5. The Parity of p(n) 61 2.6. Notes 67 Chapter 3 – Sums of Squares and Sums of Triangular Numbers 73 3.1. Lambert Series 73 3.2. Sums of Two Squares 74 3.3. Sums of Four Squares 77 3.4. Sums of Six Squares 81 3.5. Sums of Eight Squares 85 3.6. Sums of Triangular Numbers 89 3.7. Representations of Integers by x2 + 2y2, x2 + 3y2, and x2 + xy + y2 90 3.8. Notes 97 Chapter 4 – Eisenstein Series 103 4.1. Bernoulli Numbers and Eisenstein Series 103 4.2. Trigonometric Series 105 4.3. A Class of Series from Ramanujan's Lost Notebook Expressible in Terms of P, Q, and R 115 4.4. Proofs of the Congruences p(5n + 4) o 0 (mod 5) and p(7n + 5) o 0 (mod 7) 120 4.5. Notes 123 Chapter 5 – The Connection between Hypergeometric Functions and Theta Functions 127 5.1. Definitions of Hypergeometric Series and Elliptic Integrals 127 5.2. The Main Theorem 132 5.3. Principles of Duplication and Dimidiation 138 5.4. A Catalogue of Formulas for Theta Functions and Eisenstein Series 140 5.5. Notes 146 Chapter 6 – Applications of the Primary Theorem of Chapter 5 151 6.1. Introduction 151 6.2. Sums of Squares of Triangular Numbers 152 6.3. Modular Equations 158 6.4. Notes 168 Chapter 7 – The Rogers-Ramanujan Continued Fraction 169 7.1. Definition and Historical Background 171 7.2. The Convergence, Divergence, and Values of R(q) 173 7.3. The Rogers-Ramanujan Functions 176 7.4. Identities for R(q) 179 7.5. Modular Equations for R(q) 184 7.6. Notes 185 Bibliography 189 Index 203 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, Bruce 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.--jacket. Introduction -- Congruences For [partition Function] And [tau Function] -- Sums Of Squares And Sums Of Triangular Numbers -- Eisenstein Series -- The Connection Between Hypergeometric Functions And Theta Functions -- Applications Of The Primary Theorem Of Chapter 5 -- The Rogers-ramanujan Continued Fraction. Bruce C. Berndt. Includes Bibliographical References (p. 171-184) And Index. Ramanujan is recognized as one of the great number theorists of the twentieth century. This book provides an introduction to his work in number theory. It also examines subjects that have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research.