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The Riemann-Zeta Function : de Gruyter Expositions in Mathematics - A.A. Karatsuba

The Riemann-Zeta Function

de Gruyter Expositions in Mathematics

Hardcover Published: 1st August 1992
ISBN: 9783110131703
Number Of Pages: 408
For Ages: 22+ years old

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The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Industry Reviews

"[...] the scope of this well-written book is by no means restricted to the Riemann zeta-function. It spans the range successfully from elementary theory to topics of recent and current research." Mathematical Reviews

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 1p. 41
The Riemann zeta-function as a generating function in number theory
The Dirichlet series associated with the Riemann [zeta]-functionp. 43
The connection between the Riemann zeta-function and the Mobius functionp. 45
The connection between the Riemann zeta-function and the distribution of prime numbersp. 49
Explicit formulasp. 51
Prime number theoremsp. 56
The Riemann zeta-function and small sieve identitiesp. 60
Remarks on Chapter IIp. 63
Approximate functional equations
Replacing a trigonometric sum by a shorter sump. 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 derivativesp. 85
Approximate functional equation for the Hardy-Selberg function F(t)p. 95
Remarks on Chapter IIIp. 100
Vinogradov's method in the theory of the Riemann zeta-function
Vinogradov's mean value theoremp. 101
A bound for zeta sums, and some corollariesp. 112
Zero-free region for [zeta](s)p. 119
The multidimensional Dirichlet divisor problemp. 120
Remarks on Chapter IVp. 123
Density theorems
Preliminary estimatesp. 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 intervalsp. 148
Zeros of [zeta](s) in a neighborhood of the critical linep. 150
Connection between the distribution of zeros of [zeta](s) and bounds on [zeta](s) . The Lindelof conjecture and the density conjecturep. 161
Remarks on Chapter Vp. 166
Zeros of the zeta-function on the critical line
Distance between consecutive zeros on the critical linep. 168
Distance between consecutive zeros of Z[superscript k](t), k [greater than or equal to] 1p. 176
Selberg's conjecture on zeros in short intervals of the critical linep. 179
Distribution of the zeros of [zeta](s) on the critical linep. 200
Zeros of a function similar to [zeta](s) which does not satisfy the Riemann Hypothesisp. 212
Remarks on Chapter VIp. 239
Distribution of nonzero values of the Riemann zeta-function
Universality theorem for the Riemann zeta-functionp. 241
Differential independence of [zeta](s)p. 252
Distribution of nonzero values of Dirichlet L-functionsp. 255
Zeros of the zeta-functions of quadratic formsp. 272
Remarks on Chapter VIIp. 284
Behavior of [zeta]([sigma] + it), [sigma] [greater than] 1p. 286
[Omega]-theorems for [zeta](s) in the critical stripp. 290
Multidimensional [Omega]-theoremsp. 305
Remarks on Chapter VIIIp. 324
App. 1: Abel summation (partial summation)p. 326
App. 2: Some facts from analytic function theoryp. 327
App. 3: Euler's gamma-functionp. 338
App. 4: General properties of Dirichlet seriesp. 344
App. 5: Inversion formulap. 347
App. 6: Theorem on conditionally convergent series in a Hilbert spacep. 352
App. 7: Some inequalitiesp. 358
App. 8: The Kronecker and Dirichlet approximation theoremsp. 359
App. 9: Facts from elementary number theoryp. 364
App. 10: Some number theoretic inequalitiesp. 372
App. 11: Bounds for trigonometric sums (following van der Corput)p. 375
App. 12: Some algebra factsp. 380
App. 13: Gabriel's inequalityp. 381
Bibliographyp. 385
Indexp. 395
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783110131703
ISBN-10: 3110131706
Series: de Gruyter Expositions in Mathematics
Audience: General
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 408
Published: 1st August 1992
Country of Publication: DE
Dimensions (cm): 24.44 x 17.63  x 2.49
Weight (kg): 0.8