Introduction to Analytic Number Theory

Häftad, Engelska, 2012

Del 148 i serien Grundlehren der mathematischen Wissenschaften

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Beskrivning

This book has grown out of a course of lectures I have given at the Eidgenossische Technische Hochschule, Zurich. I have to acknowledge my indebtedness to Professor Carl Ludwig Siegel, who has read the book, both in manuscript and in print, and made a number of valuable criticisms and suggestions.

Produktinformation

Utgivningsdatum:2012-03-01
Mått:155 x 235 x 12 mm
Vikt:248 g
Format:Häftad
Språk:Engelska
Serie:Grundlehren der mathematischen Wissenschaften
Antal sidor:144
Förlag:Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN:9783642461262

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Innehållsförteckning

I The unique factorization theorem.- § 1. Primes.- § 2. The unique factorization theorem.- § 3. A second proof of Theorem 2.- §4. Greatest common divisor and least common multiple.- § 5. Farey sequences.- § 6. The infinitude of primes.- II Congruences.- § 1. Residue classes.- § 2. Theorems of Euler and of Fermat.- § 3. The number of solutions of a congruence.- III Rational approximation of irrationals and Hurwitz’s theorem.- § 1. Approximation of irrationals.- § 2. Sums of two squares.- § 3. Primes of the form 4k±.- §4. Hurwitz’s theorem.- IV Quadratic residues and the representation of a number as a sum of four squares.- § 1. The Legendre symbol.- § 2. Wilson’s theorem and Euler’s criterion.- § 3. Sums of two squares.- § 4. Sums of four squares.- V The law of quadratic reciprocity.- § 1. Quadratic reciprocity.- § 2. Reciprocity for generalized Gaussian sums.- § 3. Proof of quadratic reciprocity.- § 4. Some applications.- VI Arithmetical functions and lattice points.- § 1. Generalities.- § 2. The lattice point function r(n).- § 3. The divisor function d(n).- § 4. The functions ?(n).- § 5. The Möbius functions ?(n).- § 6. Euler’s function ?(n).- VII Chebyshev’s therorem on the distribution of prime numbers.- § 1. The Chebyshev functions.- § 2. Chebyshev’s theorem.- § 3. Bertrand’s postulate.- § 4. Euler’s identity.- § 5. Some formulae of Mertens.- VIII Weyl’s theorems on uniforms distribution and Kronecker’s theorem.- § 1. Introduction.- § 2. Uniform distribution in the unit interval.- § 3. Uniform distribution modulo 1.- § 4. Weyl’s theorems.- § 5. Kronecker’s theorem.- IX Minkowski’s theorem on lattice points in convex sets.- § 1. Convex sets.- § 2. Minkowski’s theorem.- § 3. Applications.- XDirichlet’s theorem on primes in an arithmetical progression.- § 1. Introduction.- § 2. Characters.- § 3. Sums of characters, orthogonality relations.- § 4. Dirichlet series, Landau’s theorem.- § 5. Dirichlet’s theorem.- XI The prime number theorem.- § 1. The non-vanishing of ? (1 + it).- § 2. The Wiener-Ikehara theorem.- § 3. The prime number theorem.- A list of books.- Notes.