9780387950976

Multiplicative Number Theory

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  • ISBN13:

    9780387950976

  • ISBN10:

    0387950974

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2000-11-01
  • Publisher: Springer Verlag
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Summary

This book thoroughly examines the distribution of prime numbers in arithmetic progressions. It covers many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetical progressions, the theorem of Siegel, and functional equations of the L-functions and their consequences for the distribution of prime numbers. In addition, a simplified, improved version of the large sieve method is presented. The 3rd edition includes a large number of revisions and corrections as well as a new section with references to more recent work in the field.

Table of Contents

Preface to the Second and Third Editions vii
Preface to the First Edition ix
Bibliography xi
Notation xiii
Primes in Arithmetic Progression
1(11)
Gauss' Sum
12(5)
Cyclotomy
17(10)
Primes in Arithmetic Progression: The General Modulus
27(8)
Primitive Characters
35(8)
Dirichlet's Class Number Formula
43(11)
The Distribution of the Primes
54(5)
Riemann's Memoir
59(6)
The Functional Equation of the L Functions
65(8)
Properties of the Γ Function
73(1)
Integral Functions of Order 1
74(5)
The Infinite Products for ξ(s) and ξ(s, x)
79(5)
A Zero-Free Region for ζ(s)
84(4)
Zero-Free Regions for L(s, χ)
88(9)
The Number N(T)
97(4)
The Number N(T, χ)
101(3)
The Explicit Formula for ψ(x)
104(7)
The Prime Number Theorem
111(4)
The Explicit Formula for ψ(x, χ)
115(6)
The Prime Number Theorem for Arithmetic Progressions (I)
121(5)
Siegel's Theorem
126(6)
The Prime Number Theorem for Arithemtic Progressions (II)
132(3)
The Polya-Vinogradov Inequality
135(3)
Further Prime Number Sums
138(5)
An Exponential Sum Formed with Primes
143(2)
Sums of Three Primes
145(6)
The Large Sieve
151(10)
Bombieri's Theorem
161(8)
An Average Result
169(3)
References to Other Work
172(3)
Index 175

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