**Student Mathematical Library**

Volume: 92;
2020;
152 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-6257-4
Product Code: STML/92**

List Price: $59.00

AMS Member Price: $47.20

MAA Member Price: $47.20

**Electronic ISBN: 978-1-4704-6279-6
Product Code: STML/92.E**

List Price: $59.00

AMS Member Price: $47.20

MAA Member Price: $47.20

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#### Supplemental Materials

# The Great Prime Number Race

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*Roger Plymen*

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.

#### Readership

Undergraduate and graduate students interested in analytic number theory.

#### Table of Contents

# Table of Contents

## The Great Prime Number Race

- Cover Cover11
- Title page iii4
- Preface ix10
- Chapter 1. The Riemann zeta function 114
- Chapter 2. The Euler product 1730
- Chapter 3. The functional equation 2740
- Chapter 4. The explicit formulas in analytic number theory 5770
- Chapter 5. The prime number theorem 8194
- Chapter 6. Oscillation of 𝜋(𝑥)-𝐿𝑖(𝑥) 93106
- Chapter 7. The prime number race 107120
- Chapter 8. Exercises, hints, and selected solutions 113126
- Bibliography 133146
- Index 137150
- Back Cover Back Cover1153