1 Preliminaries. - 1. The Problems of Representations and Their Solutions. - 2. Methods. - 3. The Contents of This Book. - 4. References. - 5. Problems. - 6. Notation. - 2 Sums of Two Squares. - 1. The One Square Problem. - 2. The Two Squares Problem. - 3. Some Early Work. - 4. The Main Theorems. - 5. Proof of Theorem 2. - 6. Proof of Theorem 3. - 7. The Circle Problem. - 8. The Determination of N2(x). - 9. Other Contributions to the Sum of Two Squares Problem. - 10. Problems. - 3 Triangular Numbers and the Representation of Integers as Sums of Four Squares. - 1. Sums of Three Squares. - 2. Three Squares Four Squares and Triangular Numbers. - 3. The Proof of Theorem 2. - 4. Main Result. - 5. Other Contributions. - 6. Proof of Theorem 4. - 7. Proof of Lemma 3. - 8. Sketch of Jacobi's Proof of Theorem 4. - 9. Problems. - 4 Representations as Sums of Three Squares. - 1. The First Theorem. - 2. Proof of Theorem 1 Part I. - 3. Early Results. - 4. Quadratic Forms. - 5. Some Needed Lemmas. -6. Proof of Theorem 1 Part II. - 7. Examples. - 8. Gauss's Theorem. - 9. From Gauss to the Twentieth Century. - 10. The Main Theorem. - 11. Some Results from Number Theory. - 12. The Equivalence of Theorem 4 with Earlier Formulations. - 13. A Sketch of the Proof of (4. 7?). - 14. Liouville's Method. - 15. The Average Order of r3(n) and the Number of Representable Integers. - 16. Problems. - 5 Legendre's Theorem. - 1. The Main Theorem and Early Results. - 2. Some Remarks and a Proof That the Conditions Are Necessary. - 3. The Hasse Principle. - 4. Proof of Sufficiency of the Conditions of Theorem 1. - 5. Problems. - 6 Representations of Integers as Sums of Nonvanishing Squares. - 1. Representations by k ? 4 Squares. - 2. Representations by k Nonvanishing Squares. - 3. Representations as Sums of Four Nonvanishing Squares. - 4. Representations as Sums of Two Nonvanishing Squares. - 5. Representations as Sums of Three Nonvanishing Squares. - 6. On the Number of Integers n ? x That Are Sumsof k Nonvanishing Squares. - 7. Problems. - 7 The Problem of the Uniqueness of Essentially Distinct Representations. - 1. The Problem. - 2. Some Preliminary Remarks. - 3. The Case k = 4. - 4. The Case k ? 5. - 5. The Cases k = 1 and k = 2. - 6. The Case k = 3. - 7. Problems. - 8 Theta Functions. - 1. Introduction. - 2. Preliminaries. - 3. Poisson Summation and Lipschitz's Formula. - 4. The Theta Functions. - 5. The Zeros of the Theta Functions. - 6. Product Formulae. - 7. Some Elliptic Functions. - 8. Addition Formulae. - 9. Problems. - 9 Representations of Integers as Sums of an Even Number of Squares. - 1. A Sketch of the Method. - 2. Lambert Series. - 3. The Computation of the Powers ?32k. - 4. Representation of Powers of ?3 by Lambert Series. - 5. Expansions of Lambert Series into Divisor Functions. - 6. The Values of the rk(n) for Even k ? 12. - 7. The Size of rk(n) for Even k ? 8. - 8. An Auxilliary Lemma. - 9. Estimate of r10(n) and r12(n). - 10. An Alternative Approach. - 11. Problems. - 10 Various Results on Representations as Sums of Squares. - 1. Some Special Older Results. - 2. More Recent Contributions. - 3. The Multiplicativity Problem. - 4. Problems. - 11 Preliminaries to the Circle Method and the Method of Modular Functions. - 1. Introduction. - 2. Farey Series. - 3. Gaussian Sums. - 4. The Modular Group and Its Subgroups. - 5. Modular Forms. - 6. Some Theorems. - 7. The Theta Functions as Modular Forais. - 8. Problems. - 12 The Circle Method. - 1. The Principle of the Method. - 2. The Evaluation of the Error Terms and Formula for rs(n). - 3. Evaluation of the Singular Series. - 4. Explicit Evaluation of L. - 5. Discussion of the Density of Representations. - 6. Other Approaches. - 7. Problems. - 13 Alternative Methods for Evaluating rs(n). - 1. Estermann's Proof. - 2. Sketch of the Proof by Modular Functions. - 3. The Function ?s(?). - 4. The Expansion of ?s(?) at the Cusp ? = -1. - 5. The Function ?s(?). - 6. Proof of Theorem 4. - 7. Modular Functions and the Number of Representations by Quadratic Forms. - 8. Problems. - 14 Recent Work. - 1. Introduction. - 2. Notation and Definitions. - 3. The Representation of Totally Positive Algebraic Integers as Sums of Squares. - 4. Some Special Results. - 5. The Circle Problem in Algebraic Number Fields. - 6. Hilbert's 17th Problem. - 7. The Work of Artin. - 8. From Artin to Pfister. - 9. The Work of Pfister and Related Work. - 10. Some Comments and Additions. - 11. Hilbert's 11th Problem. - 12. The Classification Problem and Related Topics. - 13. Quadratic Forms Over ?p. - 14. The Hasse Principle. - Appendix Open Problems. - References. - Addenda. - Author Index. Language: English