The Great Prime Number Race

Have you ever wondered about the explicit formulas in analytic number theory? This short book provides a streamlined and rigorous approach to the explicit formulas of Riemann and von Mangoldt. The race between the prime counting function and the logarithmic integral forms a motivating thread through the narrative, which emphasizes the interplay between the oscillatory terms in the Riemann formula and the Skewes number, the least number for which the prime number theorem undercounts the number of primes. Throughout the book, there are scholarly references to the pioneering work of Euler. The book includes a proof of the prime number theorem and outlines a proof of Littlewood's oscillation theorem before finishing with the current best numerical upper bounds on the Skewes number. This book is a unique text that provides all the mathematical background for understanding the Skewes number. Many exercises are included, with hints for solutions. This book is suitable for anyone with a first course in complex analysis. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians. 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 ζ(s) Chapter 2. The Euler product 2.1. The zeta function and the Euler product 2.2. The logarithmic derivative of ζ(s) 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 Li(z) 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 π(x)-Li(x) 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