Sums of Squares of Integers
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Summary
Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's elementary methods with the analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares of integers for any s and n. It is also the first book to give a proof of Szemeredi's theorem, and is the first number theory book to discuss how the modern theory of modular forms complements and clarifies the classical fundamental results about sums of squares.
The book presents several existing, yet still interesting and instructive, examples of modular forms. Two chapters develop useful properties of the Bernoulli numbers and illustrate arithmetic progressions, proving the theorems of van der Waerden, Roth, and Szemeredi. The book also explains applications of the theory to three problems that lie outside of number theory in the areas of cryptanalysis, microwave radiation, and diamond cutting. The text is complemented by the inclusion of over one hundred exercises to test the reader's understanding.
Table of Contents
Introduction
Prerequisites
Outline of Chapters 2 - 8
Elementary Methods
Introduction
Some Lemmas
Two Fundamental Identities
Euler's Recurrence for Sigma(n)
More Identities
Sums of Two Squares
Sums of Four Squares
Still More Identities
Sums of Three Squares
An Alternate Method
Sums of Polygonal Numbers
Exercises
Bernoulli Numbers
Overview
Definition of the Bernoulli Numbers
The Euler-MacLaurin Sum Formula
The Riemann Zeta Function
Signs of Bernoulli Numbers Alternate
The von Staudt-Clausen Theorem
Congruences of Voronoi and Kummer
Irregular Primes
Fractional Parts of Bernoulli Numbers
Exercises
Examples of Modular Forms
Introduction
An Example of Jacobi and Smith
An Example of Ramanujan and Mordell
An Example of Wilton: t (n) Modulo 23
An Example of Hamburger
Exercises
Hecke's Theory of Modular Forms
Introduction
Modular Group ? and its Subgroup ? 0 (N)
Fundamental Domains For ? and ? 0 (N)
Integral Modular Forms
Modular Forms of Type Mk(? 0(N);chi) and Euler-Poincare series
Hecke Operators
Dirichlet Series and Their Functional Equation
The Petersson Inner Product
The Method of Poincare Series
Fourier Coefficients of Poincare Series
A Classical Bound for the Ramanujan t function
The Eichler-Selberg Trace Formula
l-adic Representations and the Ramanujan Conjecture
Exercises
Representation of Numbers as Sums of Squares
Introduction
The Circle Method and Poincare Series
Explicit Formulas for the Singular Series
The Singular Series
Exact Formulas for the Number of Representations
Examples: Quadratic Forms and Sums of Squares
Liouville's Methods and Elliptic Modular Forms
Exercises
Arithmetic Progressions
Introduction
Van der Waerden's Theorem
Roth's Theorem t 3 = 0
Szemeredi's Proof of Roth's Theorem
Bipartite Graphs
Configurations
More Definitions
The Choice of tm
Well-Saturated K-tuples
Szemeredi's Theorem
Arithmetic Progressions of Squares
Exercises
Applications
Factoring Integers
Computing Sums of Two Squares
Computing Sums of Three Squares
Computing Sums of Four Squares
Computing rs(n)
Resonant Cavities
Diamond Cutting
Cryptanalysis of a Signature Scheme
Exercises
References
Index
Editorial Reviews
"The book provides a good introduction to theory of modular forms; a study of the chapter on Hecke's theory of modular forms prepares the reader to understand more advanced treatments…"
-CMS Notes, Vol. 38, No. 5, Sept. 2006
