VI B. C. BERND T §3.4. Sums of Six Squares 63 §3.5. Sums of Eight Squares 67 §3.6. Sums of Triangular Numbers 71 §3.7. Representations of Integers by x2 + 2y2, x2 + 3y2, and x2 + xy + y2 72 §3.8. Notes 79 Chapter 4. Eisenstein Series 85 §4.1. Bernoulli Numbers and Eisenstein Series 85 §4.2. Trigonometric Series 87 §4.3. A Class of Series from Ramanujan's Lost Notebook Expressible in Terms of P, Q, and R 97 §4.4. Proofs of the Congruences p(bn + 4) = 0 (mod 5) and p(7n + 5) = 0(mod7) 102 §4.5. Notes 105 Chapter 5. The Connection Between Hypergeometric Functions and Theta Functions 109 §5.1. Definitions of Hypergeometric Series and Elliptic Integrals 109 §5.2. The Main Theorem 114 §5.3. Principles of Duplication and Dimidiation 120 §5.4. A Catalogue of Formulas for Theta Functions and Eisenstein Series 122 §5.5. Notes 128 Chapter 6. Applications of the Primary Theorem of Chapter 5 133 §6.1. Introduction 133 §6.2. Sums of Squares and Triangular Numbers 134 §6.3. Modular Equations 140 §6.4. Notes 150 Chapter 7. The Rogers-Ramanujan Continued Fraction 153 §7.1. Definition and Historical Background 153

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