Number Theory, Fourier Analysis and Geometric Discrepancy (London Mathematical Society Student Texts, Series Number 81)

The First Part Of This Book Is Dedicated To The First Goal. The Reader Will Find Some Topics Typically Presented In Introductory Books On Number Theory: Factorization, Arithmetic Functions And Integer Points, Congruences And Cryptography, Quadratic Reciprocity, And Sums Of Two And Four Squares. Starting From The First Few Pages We Introduce Some Simple And Captivating Findings, Such As Chebyshev's Theorem And The Elementary Results For The Gauss Circle Problem And For The Dirichlet Divisor Problem, Which May Lead The Reader To A Deeper Study Of Number Theory, Particularly Students Who Are Interested In Calculus And Analysis-- Part One: Elementary Number Theory -- Prelude -- Arithmetic Functions And Integer Points -- Congruences -- Quadratic Reciprocity And Fourier Series -- Sums Of Squares -- Part Two: Fourier Analysis And Geometric Discrepancy -- Uniform Distribution And Compoleteness Of The Trigonometric System -- Discrepeancy And Trigonometric Apporximation -- Integer Points And Poisson Summation Formula -- Integer Oints And Exponetial Sums -- Geometric Discrepancy And Decay Of Fourier Transforms -- Discrepancy In High Dimension And Bessel Functions. Giancarlo Travaglini. Includes Bibliographical References And Index. The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the KoksmaHlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions. "The first part of this book is dedicated to the first goal. The reader will find some topics typically presented in introductory books on Number Theory: factorization, arithmetic functions and integer points, congruences and cryptography, quadratic reciprocity, and sums of two and four squares. Starting from the first few pages we introduce some simple and captivating findings, such as Chebyshev's theorem and the elementary results for the Gauss circle problem and for the Dirichlet divisor problem, which may lead the reader to a deeper study of Number Theory, particularly students who are interested in Calculus and Analysis"-- Provided by publisher The first part of this book is dedicated to the first goal. The reader will find some topics typically presented in introductory books on Number Theory: factorization, arithmetic functions and integer points, congruences and cryptography, quadratic reciprocity, and sums of two and four squares. Starting from the first few pages we introduce some simple and captivating findings, such as Chebyshev's theorem and the elementary results for the Gauss circle problem and for the Dirichlet divisor problem, which may lead the reader to a deeper study of Number Theory, particularly students who are interested in Calculus and Analysis.--Résumé de l'éditeur Cover 1 Title: Number Theory, Fourier Analysis and Geometric Discrepancy 2 Contents 10 Introduction 12 1 Prelude 14 2 Arithmetic functions and integer points 30 3 Congruences 52 4 Quadratic reciprocity and Fourier series 76 5 Sums of squares 98 6 Uniform distribution and completeness of the trigonometric system 116 7 Discrepancy and trigonometric approximation 143 8 Integer points and Poisson summation formula 159 9 Integer points and exponential sums 192 10 Geometric discrepancy and decay of Fourier transforms 202 11 Discrepancy in high dimension and Bessel functions 223 References 238 Index 247