Geometric and Analytic Number Theory (Universitext) de Edmund Hlawka

Geometric and Analytic Number Theory: Universitext

Edmund Hlawka

Charles Thomas

Johannes Schoißengeier

Rudolf Taschner

In the English edition, the chapter on the Geometry of Numbers has been enlarged to include the important findings of H. Lenstraj furthermore, tried and tested examples and exercises have been included. The translator, Prof. Charles Thomas, has solved the difficult problem of the German text into English in an admirable way. He deserves transferring our 'Unreserved praise and special thailks. Finally, we would like to express our gratitude to Springer-Verlag, for their commitment to the publication of this English edition, and for the special care taken in its production. Vienna, March 1991 E. Hlawka J. SchoiBengeier R. Taschner Preface to the German Edition We have set ourselves two aims with the present book on number theory. On the one hand for a reader who has studied elementary number theory, and who has knowledge of analytic geometry, differential and integral calculus, together with the elements of complex variable theory, we wish to introduce basic results from the areas of the geometry of numbers, diophantine ap proximation, prime number theory, and the asymptotic calculation of number theoretic functions. However on the other hand for the student who has al ready studied analytic number theory, we also present results and principles of proof, which until now have barely if at all appeared in text books.

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Cuprins

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.