The Riemann Zeta-Function

Anatoly A. Karatsuba, S. M. Voronin, Neal Koblitz

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110131703
• eBook ISBN: 9783110886146
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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Table of contents

• 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
• §2. The connection between the Riemann zeta-function and the Möbius function
• §3. 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
• §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
• §6. Connection between the distribution of zeros of ζ(s) and bounds on |ζ(s)|. The Lindelöf conjecture and the density conjecture
• Remarks 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
• Chapter VII. Distribution of nonzero values of the Riemann zeta-function
• §1. 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
• §4. General properties of Dirichlet series
• §5. Inversion formula
• §6. Theorem on conditionally convergent series in a Hilbert space
• §7. Some inequalities
• §8. The Kronecker and Dirichlet approximation theorems
• §9. Facts from elementary number theory
• §10. Some number theoretic inequalities
• §11. Bounds for trigonometric sums (following van der Corput)
• §12. Some algebra facts
• §13. Gabriel’s inequality
• Bibliography
• Index