One form of the Prime Number Theorem is that \( \pi(x) \), the number of primes less than or equal to \( x \), is asymptotically equal to the logarithm integral function \( \mathrm{Li}(x) \), defined by

\( \mathrm{Li}(x) = \int_{0}^{x} \frac{dt}{\ln t}, \)

so that we have \( \pi(x)/\mathrm{Li}(x) \rightarrow 1 \) as \( x \rightarrow \infty \). Experimentally \( \mathrm{Li}(x) \) is always high, that is, \( \mathrm{Li}(x) - \pi(x) \) is always positive, but J. E. Littlewood proved in 1914 that this difference changes signs infinitely often. If we think of this as a race to infinity between \( \pi(x) \) and \( \mathrm{Li}(x) \), then each one pulls ahead of the other infinitely often, hence the book’s title. A still unanswered question is the \( x \) at which \( \pi(x) \) first pulls ahead, which this book denotes \( \Xi \) and calls Skewes’s number. Stanley Skewes in 1933 (assuming the Riemann Hypothesis) and in 1955 (unconditionally) was the first to give an explicit upper bound for \( \Xi \). Many authors use “Skewes’s number” to mean Skewes’s upper bound instead of \( \Xi \). The upper and lower bounds have been improved since then, and the best-known values are \( 19 \ln 10 < \ln \Xi < 727.951346801\).

This book is a discursive introduction to the analytic theory of prime numbers, steering us toward Littlewood’s theorem but with many side paths. It is aimed at upper-level undergraduates, and the material was used by the author for a series of lectures at that level. It assumes a previous course in complex variables (to include such topics as the Weierstrass factorization theorem), but otherwise starts at the beginning.

The main line of the exposition is very traditional, traveling through the Riemann zeta function, its analytic continuation, its functional equation, the explicit formulas, and the oscillation theorems. This list is essentially the Table of Contents of Ingham’s 1932 The Distribution of Prime Numbers. It also has some interesting sidelines, such as the question “Can you hear the Riemann Hypothesis” (by translating the zeroes into sounds) and a new class of numbers developed by Maxim Kontsevich and Don Zagier called “periods”, that contains the algebraic numbers and most interesting transcendental numbers but is still countable. The book concludes with a sketch of Littlewood’s proof and of R. S. Lehman’s 1966 improved proof that gives an effective upper bound for \( \Xi \), with a lot of qualitative information about why they work.

The book includes an exercises section at the end, with hints and solutions. Most of the exercises are not very difficult, but are about the right level for the undergraduates this book is aimed at. There are a large number of prime number races in number theory, many of which concern the number of primes in different congruence classes. For example, empirically there are always more primes less than \( x \) of the form \( 4n + 3 \) than of the form \( 4n + 1\), but Hardy and Littlewood in 1918 proved by the same method as Littlewood’s theorem that the lead between these two also changes places infinitely often.