Contents

Preface ix

Chapter 1. Introduction 1

§1.1. Notation and Arithmetical Functions 1

§1.2. What are ^-Series and Theta Functions? 6

§1.3. Fundamental Theorems about g-Series and Theta

Functions 7

§1.4. Notes 22

Chapter 2. Congruences for p(n) and r(n) 27

§2.1. Historical Background 27

§2.2. Elementary Congruences for r{n) 28

§2.3. Ramanujan's Congruence p(5n -f 4) = 0 (mod 5) 31

§2.4. Ramanujan's Congruence p(7n + 5) = 0 (mod 7) 39

§2.5. The Parity of p(n) 43

§2.6. Notes 49

Chapter 3. Sums of Squares and Sums of Triangular Numbers 55

§3.1. Lambert Series 55

§3.2. Sums of Two Squares 56

§3.3. Sums of Four Squares 59