UID:
almahu_9947360001002882
Format:
Online-Ressource (xii, 396 p)
Edition:
Online-Ausg. Palo Alto, Calif ebrary 2011 Electronic reproduction; Available via World Wide Web
ISBN:
3110131706 (cloth : acid-free)
,
9783110131703 (cloth : acid-free)
Series Statement:
De Gruyter expositions in mathematics 5
Content:
The Riemann Zeta-Function (De Gruyter Expositions in Mathematics)
Note:
Includes bibliographical references and index
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2. The connection between the Riemann zeta-function and the Möbius function3. The connection between the Riemann zeta-function and the distribution of prime numbers; 4. Explicit formulas; 5. Prime number theorems; 6. The Riemann zeta-function and small sieve identities; Remarks on Chapter II; Chapter III. Approximate functional equations; 1. Replacing a trigonometric sum by a shorter sum; 2. A simple approximate functional equation for ζ (s, α); 3. Approximate functional equation for ζ(s); 4. Approximate functional equation for the Hardy function Z(t) and its derivatives.
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4. General properties of Dirichlet series.
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5. Approximate functional equation for the Hardy-Selberg function F(t)Remarks on Chapter III; Chapter IV. Vinogradov's method in the theory of the Riemann zeta-function; 1. Vinogradov's mean value theorem; 2. A bound for zeta sums, and some corollaries; 3. Zero-free region for ζ (s); 4. The multidimensional Dirichlet divisor problem; Remarks on Chapter IV; Chapter V. Density theorems; 1. Preliminary estimates; 2. A simple bound for Ν(σ, Τ); 3. A modern estimate for Ν(σ, Τ); 4. Density theorems and primes in short intervals; 5. Zeros of ζ (s) in a neighborhood of the critical line.
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6. Connection between the distribution of zeros of ζ(s) and bounds on |ζ(s)|. The Lindelöf conjecture and the density conjectureRemarks on Chapter V; Chapter VI. Zeros of the zeta-function on the critical line; 1. Distance between consecutive zeros on the critical line; 2. Distance between consecutive zeros of Z(k)(t), k ≥ 1; 3. Selberg's conjecture on zeros in short intervals of the critical line; 4. Distribution of the zeros of on the critical line; 5. Zeros of a function similar to ζ(s) which does not satisfy the Riemann Hypothesis; Remarks on Chapter VI.
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Chapter VII. Distribution of nonzero values of the Riemann zeta-function1. Universality theorem for the Riemann zeta-function; 2. Differential independence of; 3. Distribution of nonzero values of Dirichlet L-functions; 4. Zeros of the zeta-functions of quadratic forms; Remarks on Chapter VII; Chapter VIII. Ω-theorems; 1. Behavior of |ζ(σ + it)|, σ › I; 2. Ω-theorems for ζ(s) in the critical strip; 3. Multidimensional Ω-theorems; Remarks on Chapter VIII; Appendix; 1. Abel summation (partial summation); 2. Some facts from analytic function theory; 3. Euler's gamma-function.
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Preface; Notation; Introduction; Chapter I. The definition and the simplest properties of the Riemann zeta-function; 1. Definition of ζ (s); 2. Generalizations of ζ (s); 3. The functional equation of ζ (s); 4. Functional Equations for L(s, χ) and ζ (s, α); 5. Weierstrass product for ζ(s) and L(s, χ); 6. The simplest theorems concerning the zeros of ζ (s); 7. The simplest theorems concerning the zeros of L(s, χ); Remarks on Chapter I; Chapter II. The Riemann zeta-function as a generating function in number theory; 1. The Dirichlet series associated with the Riemann ζ-function.
Additional Edition:
ISBN 9783110886146
Language:
English
DOI:
10.1515/9783110886146
URL:
http://www.degruyter.com/doi/book/10.1515/9783110886146
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