Geometric and Analytic Number Theory (Universitext) - Brossura

Hlawka, Edmund

Brossura

ISBN 10: 3540520163 ISBN 13: 9783540520160

Casa editrice: Springer-Verlag, 1991

Riassunto

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

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Contenuti

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 Pólya 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 Lovász.- Addenda.- Solutions for the Exercises.- Index of Names.- Index of Terms.

Dettagli dell'edizione

• Editore: Springer-Verlag
• Data di pubblicazione: 1991
• Lingua: Inglese
• ISBN 10: 3540520163
• ISBN 13: 9783540520160
• Rilegatura: Copertina flessibile
• Numero di pagine: 252
• Contatto del produttore: non disponibile
• Persona responsabile: CMN Printpool Inh. Mathis Neumann; https://www.also.com/ec/cms5/en_6000/6000/index.jsp; ; Friedrich-Karl-Str. 9; Hamburg; Germania