The Great Prime Number Race

Cover
Title page
Preface
Chapter 1. The Riemann zeta function
1.1. Introduction
1.2. The Riemann zeta function
1.3. The prime numbers
1.4. The Riemann zeta function
1.5. Euler and the zeta function
1.6. Meromorphic continuation of ?(?)
Chapter 2. The Euler product
2.1. The zeta function and the Euler product
2.2. The logarithmic derivative of ?(?)
Chapter 3. The functional equation
3.1. The gamma function
3.2. The functional equation
3.3. Some zeta values
3.4. Euler and the functional equation
3.5. The Euler constant revisited
Chapter 4. The explicit formulas in analytic number theory
4.1. The von Mangoldt explicit formula
4.2. Can you hear the Riemann hypothesis?
4.3. Comparison with Fourier series
4.4. Proof of the von Mangoldt formula
4.5. The logarithmic integral ??(?)
4.6. The Riemann formula
4.7. Origin of the Riemann explicit formula
Chapter 5. The prime number theorem
5.1. The Riemann-Ramanujan approximation
5.2. Proof of the prime number theorem
Chapter 6. Oscillation of ?(?)-??(?)
6.1. Littlewood’s theorem
6.2. Lehman’s theorem
Chapter 7. The prime number race
7.1. On the logarithmic density
7.2. Upper bounds for the Skewes number
Chapter 8. Exercises, hints, and selected solutions
8.1. Exercises
8.2. Hints and selected solutions
Bibliography
Index
Back Cover