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AuthorGrosswald, Emil. author
TitleRepresentations of Integers as Sums of Squares [electronic resource] / by Emil Grosswald
ImprintNew York, NY : Springer New York, 1985
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Descript 251p. online resource


During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but conยญ sisted of particularly gifted and eager listeners; unfortunately, their backยญ ground varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the intenยญ tion of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be


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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