Representations of Integers as Sums of Squares

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.