# Book:Aleksandar Ivić/The Riemann Zeta-Function

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

Published $\text {1985}$, Dover Publications, Inc

ISBN 0-486-42813-3

### Contents

PREFACE
NOTATION
ERRATA
1. ELEMENTARY THEORY
1.1. Definition of $\map \zeta s$ and Elementary Properties
1.2. The Functional Equation
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 $\map \zeta s$
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 $\map \zeta s$
4.2. The Approximate Functional Equation for $\map {\zeta^2} s$
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
6.3. Estimation of the Zeta Sum
6.4. The Order Estimate of $\map \zeta s$ 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 $\map \zeta s$ 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 $\map \psi x$
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 $\map {\Delta_2} x$ and $\map {\Delta_3} x$
13.3. Estimates of $\map {\Delta_k} x$ by Power Moments of the Zeta-Function
13.4. Estimates of $\map {\Delta_k} x$ When $k$ Is Very Large
13.5. Estimates of $\beta_k$
13.6. Mean-square Estimates of $\map {\Delta_k} x$
13.7. Large Values and Power Moments of $\map {\Delta_k} x$
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 $\map {d_z} n$
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 $\map E t$
15.5. The Connection Between $\map E T$ and $\map \Delta x$
15.6. Large Values and Power Moments of $\map E T$
APPENDIX
REFERENCES
AUTHOR INDEX
SUBJECT INDEX

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