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