Book:Aleksandar Ivić/The Riemann Zeta-Function

Aleksandar Ivić: The Riemann Zeta-Function: Theory and Applications

Published $1985$, Dover Publications, Inc

ISBN 0-486-42813-3.

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

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