The Riemann Zeta-Function

by Anatoly A. Karatsuba; S. M. Voronin

• ISBN13: 9783110131703
• ISBN10: 3110131706
• Format: Nonspecific Binding
• Copyright: 2011-05-03
• Publisher: De Gruyter

Table of Contents

Preface
Notation
Introduction
The definition and the simplest properties of the Riemann zeta-function
Definition of [zeta](s)	p. 1
Generalizations of [zeta](s)	p. 3
The functional equation of [zeta](s)	p. 5
Functional Equations for L(s, [chi]) and [zeta](s, [alpha])	p. 11
Weierstrass product for [zeta](s) and L(s, [chi])	p. 20
The simplest theorems concerning the zeros of [zeta](s)	p. 21
The simplest theorems concerning the zeros of L(s, [chi])	p. 28
Asymptotic formula for N(T)	p. 39
Remarks on Chapter 1	p. 41
The Riemann zeta-function as a generating function in number theory
The Dirichlet series associated with the Riemann [zeta]-function	p. 43
The connection between the Riemann zeta-function and the Mobius function	p. 45
The connection between the Riemann zeta-function and the distribution of prime numbers	p. 49
Explicit formulas	p. 51
Prime number theorems	p. 56
The Riemann zeta-function and small sieve identities	p. 60
Remarks on Chapter II	p. 63
Approximate functional equations
Replacing a trigonometric sum by a shorter sum	p. 64
A simple approximate functional equation for [zeta](s, [alpha])	p. 78
Approximate functional equation for [zeta](s)	p. 81
Approximate functional equation for the Hardy function Z(t) and its derivatives	p. 85
Approximate functional equation for the Hardy-Selberg function F(t)	p. 95
Remarks on Chapter III	p. 100
Vinogradov's method in the theory of the Riemann zeta-function
Vinogradov's mean value theorem	p. 101
A bound for zeta sums, and some corollaries	p. 112
Zero-free region for [zeta](s)	p. 119
The multidimensional Dirichlet divisor problem	p. 120
Remarks on Chapter IV	p. 123
Density theorems
Preliminary estimates	p. 126
A simple bound for N([sigma], T)	p. 128
A modern estimate for N([sigma], T)	p. 131
Density theorems and primes in short intervals	p. 148
Zeros of [zeta](s) in a neighborhood of the critical line	p. 150
Connection between the distribution of zeros of [zeta](s) and bounds on [zeta](s) . The Lindelof conjecture and the density conjecture	p. 161
Remarks on Chapter V	p. 166
Zeros of the zeta-function on the critical line
Distance between consecutive zeros on the critical line	p. 168
Distance between consecutive zeros of Z[superscript k](t), k [greater than or equal to] 1	p. 176
Selberg's conjecture on zeros in short intervals of the critical line	p. 179
Distribution of the zeros of [zeta](s) on the critical line	p. 200
Zeros of a function similar to [zeta](s) which does not satisfy the Riemann Hypothesis	p. 212
Remarks on Chapter VI	p. 239
Distribution of nonzero values of the Riemann zeta-function
Universality theorem for the Riemann zeta-function	p. 241
Differential independence of [zeta](s)	p. 252
Distribution of nonzero values of Dirichlet L-functions	p. 255
Zeros of the zeta-functions of quadratic forms	p. 272
Remarks on Chapter VII	p. 284
[Omega]-theorems
Behavior of [zeta]([sigma] + it), [sigma] [greater than] 1	p. 286
[Omega]-theorems for [zeta](s) in the critical strip	p. 290
Multidimensional [Omega]-theorems	p. 305
Remarks on Chapter VIII	p. 324
App. 1: Abel summation (partial summation)	p. 326
App. 2: Some facts from analytic function theory	p. 327
App. 3: Euler's gamma-function	p. 338
App. 4: General properties of Dirichlet series	p. 344
App. 5: Inversion formula	p. 347
App. 6: Theorem on conditionally convergent series in a Hilbert space	p. 352
App. 7: Some inequalities	p. 358
App. 8: The Kronecker and Dirichlet approximation theorems	p. 359
App. 9: Facts from elementary number theory	p. 364
App. 10: Some number theoretic inequalities	p. 372
App. 11: Bounds for trigonometric sums (following van der Corput)	p. 375
App. 12: Some algebra facts	p. 380
App. 13: Gabriel's inequality	p. 381
Bibliography	p. 385
Index	p. 395
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