Geometric and Analytic Number Theory

Abstract

1. The Dirichlet Approximation Theorem.- Dirichlet approximation theorem - Elementary number theory - Pell equation - Cantor series - Irrationality of ?(2) and ?(3) - multidimensional diophantine approximation - Siegel's lemma - Exercises on Chapter 1..- 2. The Kronecker Approximation Theorem.- Reduction modulo 1 - Comments on Kronecker's theorem - Linearly independent numbers - Estermann's proof - Uniform Distribution modulo 1 - Weyl's criterion - Fundamental equation of van der Corput - Main theorem of uniform distribution theory - Exercises on Chapter 2..- 3. Geometry of Numbers.- Lattices - Lattice constants - Figure lattices - Fundamental region - Minkowski's lattice point theorem - Minkowski's linear form theorem - Product theorem for homogeneous linear forms - Applications to diophantine approximation - Lagrange's theorem - the lattice?(i) - Sums of two squares - Blichfeldt's theorem - Minkowski's and Hlawka's theorem - Rogers' proof - Exercises on Chapter 3..- 4. Number Theoretic Functions.- Landau symbols - Estimates of number theoretic functions - Abel transformation - Euler's sum formula - Dirichlet divisor problem - Gauss circle problem - Square-free and k-free numbers - Vinogradov's lemma - Formal Dirichlet series - Mangoldt's function - Convergence of Dirichlet series - Convergence abscissa - Analytic continuation of the zeta- function - Landau's theorem - Exercises on Chapter 4..- 5. The Prime Number Theorem.- Elementary estimates - Chebyshev's theorem - Mertens' theorem - Euler's proof of the infinity of prime numbers - Tauberian theorem of Ingham and Newman - Simplified version of the Wiener-Ikehara theorem - Mertens' trick - Prime number theorem - The ?-function for number theory in ?(i) - Hecke's prime number theorem for ?(i) - Exercises on Chapter 5..- 6. Characters of Groups of Residues.- Structure of finite abelian groups - The character group - Dirichlet characters - Dirichlet L-series - Prime number theorem for arithmetic progressions - Gauss sums - Primitive characters - Theorem of Polya and Vinogradov - Number of power residues - Estimate of the smallest primitive root - Quadratic reciprocity theorem - Quadratic Gauss sums - Sign of a Gauss sum - Exercises on Chapter 6..- 7. The Algorithm of Lenstra, Lenstra and Lovasz.- Addenda.- Solutions for the Exercises.- Index of Names.- Index of Terms.