Book:Aleksandar Ivić/The Riemann Zeta-Function

Subject Matter

 * Riemann Zeta Function

Contents

 * PREFACE


 * NOTATION


 * ERRATA


 * 1. ELEMENTARY THEORY
 * 1.1. Definition of $\zeta \left({s}\right)$ and Elementary Properties
 * 1.2. The Functional Equation
 * 1.3. The Hadamard Product Formula
 * 1.4. The Riemann-von Mangoldt Formula
 * 1.5. An Approximate Functional Equation
 * 1.6. Mean Value Theorems
 * 1.7. Various Dirichlet Series Connected with $\zeta \left({s}\right)$
 * 1.8. Other Zeta-Functions
 * 1.9. Unproved Hypotheses


 * 2. EXPONENTIAL INTEGRALS AND EXPONENTIAL SUMS
 * 2.1. Exponential Integrals
 * 2.2. Exponential Sums
 * 2.3. The Theory of Exponent Pairs
 * 2.4. Two-Dimensional Exponent Pairs


 * 3. THE VORONOI SUMMATION FORMULA
 * 3.1. Introduction
 * 3.2. The Truncated Voronoi Formula
 * 3.3. The Weighted Voronoi Formulas
 * 3.4. Other Formulas of the Voronoi Type


 * 4. THE APPROXIMATE FUNCTIONAL EQUATIONS
 * 4.1. The Approximate Functional Equation for $\zeta \left({s}\right)$
 * 4.2. The Approximate Functional Equation for $\zeta^2 \left({s}\right)$
 * 4.3. The Approximate Functional Equation for Higher Powers
 * 4.4. The Reflection Principle


 * 5. THE FOURTH POWER MOMENT
 * 5.1. Introduction
 * 5.2. The Mean Value Theorem for Dirichlet Polynomials
 * 5.3. Proof of the Fourth Power Moment Estimate


 * 6. THE ZERO-FREE REGION
 * 6.1. A Survey of Results
 * 6.2. The Method of Vinogradov-Korobov
 * 6.3. Estimation of the Zeta Sum
 * 6.4. The Order Estimate of $\zeta \left({s}\right)$ Near $\sigma = 1$
 * 6.5. The Deduction of the Zero-Free Region


 * 7. MEAN VALUE ESTIMATES OVER SHORT INTERVALS
 * 7.1. Introduction
 * 7.2. An Auxiliary Estimate
 * 7.3. The Mean Square When $\sigma$ Is in the Critical Strip
 * 7.4. The Mean Square When $\sigma = \tfrac 1 2$
 * 7.5. The Order of $\zeta \left({s}\right)$ in the Critical Strip
 * 7.6. Third and Fourth Power Moments in Short Intervals


 * 8. HIGHER POWER MOMENTS
 * 8.1. Introduction
 * 8.2. Some Convexity Estimates
 * 8.3. Power Moments for $\sigma = \tfrac 1 2$
 * 8.4. Power Moments for $\tfrac 1 2 < \sigma < 1$
 * 8.5. Asymptotic Formulas for Power Moments When $\tfrac 1 2 < \sigma < 1$


 * 9. OMEGA RESULTS
 * 9.1. Introduction,
 * 9.2. Omega Results When $\sigma \ge 1$
 * 9.3. Lemmas on Certain Order Results
 * 9.4. Omega Results for $\tfrac 1 2 \le \sigma \le 1$
 * 9.5. Lower Bounds for Power Moments When $\sigma = \tfrac 1 2$


 * 10. ZEROS ON THE CRITICAL LINE
 * 10.1. Levinson's Method
 * 10.2. Zeros on the Critical Line in Short Intervals
 * 10.3. Consecutive Zeros on the Critical Line


 * 11. ZERO-DENSITY ESTIMATES
 * 11.1. Introduction
 * 11.2. The Zero-Detection Method
 * 11.3. The Ingham-Huxley Estimates
 * 11.4. Estimates for $\sigma$ Near Unity
 * 11.5. Reflection Principle Estimates
 * 11.6. Double Zeta Sums
 * 11.7. Zero-Density Estimates for $\tfrac 3 4 < \sigma < 1$
 * 11.8. Zero-Density Estimates for $\sigma$ Close to $\tfrac 3 4$


 * 12. THE DISTRIBUTION OF PRIMES
 * 12.1. General Remarks
 * 12.2. The Explicit Formula for $\psi \left({x}\right)$
 * 12.3. The Prime Number Theorem
 * 12.4. The Generalised von Mangoldt Function and the Möbius Function
 * 12.5. Von Mangoldt's Function in Short Intervals
 * 12.6. The Difference between Consecutive Primes
 * 12.7. Almost Primes in Short Intervals
 * 12.8. Sums of Differences between Consecutive Primes


 * 13. THE DIRICHLET DIVISOR PROBLEM
 * 13.1. Introduction
 * 13.2. Estimates for $\Delta_2 \left({x}\right)$ and $\Delta_3 \left({x}\right)$
 * 13.3. Estimates of $\Delta_k \left({x}\right)$ by Power Moments of the Zeta-Function
 * 13.4. Estimates of $\Delta_k \left({x}\right)$ When $k$ Is Very Large
 * 13.5. Estimates of $\beta_k$
 * 13.6. Mean-square Estimates of $\Delta_k \left({x}\right)$
 * 13.7. Large Values and Power Moments of $\Delta_k \left({x}\right)$
 * 13.8. The Circle Problem


 * 14. VARIOUS OTHER DIVISOR PROBLEMS
 * 14.1. Summatory Functions of Arithmetical Convolutions
 * 14.2. Some Applications of the Convolution Method
 * 14.3. Three-Dimensional Divisor Problems
 * 14.4. Powerful Numbers
 * 14.5. Nonisomorphic Abelian Groups of a Given Order
 * 14.6. The General Divisor Function $d_z \left({n}\right)$
 * 14.7. Small Additive Functions


 * 15. ATKINSON'S FORMULA FOR THE MEAN SQUARE
 * 15.1. Introduction
 * 15.2. Proof of Atkinson's Formula
 * 15.3. Modified Atkinson's Formula
 * 15.4. The Mean Square of $E \left({t}\right)$
 * 15.5. The Connection Between $E \left({T}\right)$ and $\Delta \left({x}\right)$
 * 15.6. Large Values and Power Moments of $E \left({T}\right)$


 * APPENDIX


 * REFERENCES


 * AUTHOR INDEX


 * SUBJECT INDEX