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Geometric and Analytic Number Theory : Universitext - Edmund Hlawka

Geometric and Analytic Number Theory



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Based on lectures given by Professor Hlawka in Vienna and in Pasadena, the book covers diophantine approximation, uniform distribution of numbers modulo 1 and geometry of numbers, as well as analytic number theory. It also proves the irrationality of zeta (3) and presents the important method of A. and H. Lenstra for the decomposition of polynomials, neither available in any other textbook. Various proofs of the prime number theorem are described. This monograph on number theory, real analysis, algebra/algebraic number theory is intended for students and lecturers in the above fields.

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.

ISBN: 9783540520160
ISBN-10: 3540520163
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 238
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.13 x 17.02  x 1.27
Weight (kg): 0.41