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Author Grosswald, Emil. author Representations of Integers as Sums of Squares [electronic resource] / by Emil Grosswald New York, NY : Springer New York, 1985 http://dx.doi.org/10.1007/978-1-4613-8566-0 251p. online resource

SUMMARY

CONTENT

1 Preliminaries -- ยง1. The Problems of Representations and Their Solutions -- ยง2. Methods -- ยง3. The Contents of This Book -- ยง4. References -- ยง5. Problems -- ยง6. Notation -- 2 Sums of Two Squares -- ยง1. The One Square Problem -- ยง2. The Two Squares Problem -- ยง3. Some Early Work -- ยง4. The Main Theorems -- ยง5. Proof of Theorem 2 -- ยง6. Proof of Theorem 3 -- ยง7. The โ{128}{156}Circle Problemโ{128}{157} -- ยง8. The Determination of N2(x) -- ยง9. Other Contributions to the Sum of Two Squares Problem -- ยง10. Problems -- 3 Triangular Numbers and the Representation of Integers as Sums of Four Squares -- ยง1. Sums of Three Squares -- ยง2. Three Squares, Four Squares, and Triangular Numbers -- ยง3. The Proof of Theorem 2 -- ยง4. Main Result -- ยง5. Other Contributions -- ยง6. Proof of Theorem 4 -- ยง7. Proof of Lemma 3 -- ยง8. Sketch of Jacobiโ{128}{153}s Proof of Theorem 4 -- ยง9. Problems -- 4 Representations as Sums of Three Squares -- ยง1. The First Theorem -- ยง2. Proof of Theorem 1, Part I -- ยง3. Early Results -- ยง4. Quadratic Forms -- ยง5. Some Needed Lemmas -- ยง6. Proof of Theorem 1, Part II -- ยง7. Examples -- ยง8. Gaussโ{128}{153}s Theorem -- ยง9. From Gauss to the Twentieth Century -- ยง10. The Main Theorem -- ยง11. Some Results from Number Theory -- ยง12. The Equivalence of Theorem 4 with Earlier Formulations -- ยง13. A Sketch of the Proof of (4.7?) -- ยง14. Liouvilleโ{128}{153}s Method -- ยง15. The Average Order of r3(n) and the Number of Representable Integers -- ยง16. Problems -- 5 Legendreโ{128}{153}s Theorem -- ยง1. The Main Theorem and Early Results -- ยง2. Some Remarks and a Proof That the Conditions Are Necessary -- ยง3. The Hasse Principle -- ยง4. Proof of Sufficiency of the Conditions of Theorem 1 -- ยง5. Problems -- 6 Representations of Integers as Sums of Nonvanishing Squares -- ยง1. Representations by k ? 4 Squares -- ยง2. Representations by k Nonvanishing Squares -- ยง3. Representations as Sums of Four Nonvanishing Squares -- ยง4. Representations as Sums of Two Nonvanishing Squares -- ยง5. Representations as Sums of Three Nonvanishing Squares -- ยง6. On the Number of Integers n ? x That Are Sums of k Nonvanishing Squares -- ยง7. Problems -- 7 The Problem of the Uniqueness of Essentially Distinct Representations -- ยง1. The Problem -- ยง2. Some Preliminary Remarks -- ยง3. The Case k = 4 -- ยง4. The Case k ? 5 -- ยง5. The Cases k = 1 and k = 2 -- ยง6. The Case k = 3 -- ยง7. Problems -- 8 Theta Functions -- ยง1. Introduction -- ยง2. Preliminaries -- ยง3. Poisson Summation and Lipschitzโ{128}{153}s Formula -- ยง4. The Theta Functions -- ยง5. The Zeros of the Theta Functions -- ยง6. Product Formulae -- ยง7. Some Elliptic Functions -- ยง8. Addition Formulae -- ยง9. Problems -- 9 Representations of Integers as Sums of an Even Number of Squares -- ยง1. A Sketch of the Method -- ยง2. Lambert Series -- ยง3. The Computation of the Powers ?32k -- ยง4. Representation of Powers of ?3 by Lambert Series -- ยง5. Expansions of Lambert Series into Divisor Functions -- ยง6. The Values of the rk(n) for Even k ? 12 -- ยง7. The Size of rk(n) for Even k ? 8 -- ยง8. An Auxilliary Lemma -- ยง9. Estimate of r10(n) and r12(n) -- ยง10. An Alternative Approach -- ยง11. Problems -- 10 Various Results on Representations as Sums of Squares -- ยง1. Some Special, Older Results -- ยง2. More Recent Contributions -- ยง3. The Multiplicativity Problem -- ยง4. Problems -- 11 Preliminaries to the Circle Method and the Method of Modular Functions -- ยง1. Introduction -- ยง2. Farey Series -- ยง3. Gaussian Sums -- ยง4. The Modular Group and Its Subgroups -- ยง5. Modular Forms -- ยง6. Some Theorems -- ยง7. The Theta Functions as Modular Forais -- ยง8. Problems -- 12 The Circle Method -- ยง1. The Principle of the Method -- ยง2. The Evaluation of the Error Terms and Formula for rs(n) -- ยง3. Evaluation of the Singular Series -- ยง4. Explicit Evaluation of L -- ยง5. Discussion of the Density of Representations -- ยง6. Other Approaches -- ยง7. Problems -- 13 Alternative Methods for Evaluating rs(n) -- ยง1. Estermannโ{128}{153}s Proof -- ยง2. Sketch of the Proof by Modular Functions -- ยง3. The Function ?s(?) -- ยง4. The Expansion of ?s(?) at the Cusp ? = -1 -- ยง5. The Function ?s(?) -- ยง6. Proof of Theorem 4 -- ยง7. Modular Functions and the Number of Representations by Quadratic Forms -- ยง8. Problems -- 14 Recent Work -- ยง1. Introduction -- ยง2. Notation and Definitions -- ยง3. The Representation of Totally Positive Algebraic Integers as Sums of Squares -- ยง4. Some Special Results -- ยง5. The Circle Problem in Algebraic Number Fields -- ยง6. Hilbertโ{128}{153}s 17th Problem -- ยง7. The Work of Artin -- ยง8. From Artin to Pfister -- ยง9. The Work of Pfister and Related Work -- ยง10. Some Comments and Additions -- ยง11. Hilbertโ{128}{153}s 11th Problem -- ยง12. The Classification Problem and Related Topics -- ยง13. Quadratic Forms Over ?p -- ยง14. The Hasse Principle -- Appendix Open Problems -- References -- Addenda -- Author Index

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