Cambridge Studies in Advanced Mathematics 97. Cambridge: Cambridge University Press (ISBN 0-521-84903-9/hbk). xvii, 552 p. £ 48.00; $ 90.00; $ 72.00/e-book (2007).

This book gives an introduction to the classical analytic number theory. The topics covered are elementary estimates for prime numbers and arithmetic functions, the prime number theorem and the prime number theorem for arithmetic progressions with applications, Selberg’s sieve, explicit formulae and results subject to the Riemann Hypothesis, the vertical distribution of zeros, and Omega-theorems. The elementary methods presented include the Dirichlet divisor problem, Chebyshev’s bounds for $\pi(x)$, the normal order of $\omega(n)$, the number of distinct prime divisors of $n$, and the distribution of $\Omega(n)-\omega(n)$, where $\Omega(n)$ denotes the number of prime divisors counted with multiplicity. The proof of the prime number theorem via complex integration is given in great detail, a proof using the Wiener-Ikehara Tauberian theorem and an elementary proof using Selberg’s estimate for $\sum_{n\leq x}\Lambda(n)\log n + \sum_{nm\leq x}\Lambda(n)\Lambda(m)$ are also given. Selberg’s asymptotic formula for the number of integers $n\leq x$ satisfying $\omega(n)=k$ is given as a different example of complex integration. As applications of the prime number theorem the number of integers composed of small prime factors only as well as the number of integers without small prime factors are computed. Rankin’s lower bound for the maximal difference between consecutive prime numbers is given, as is the theorem by Hensley and Richards that the prime $k$-tuple conjecture and the conjecture $\pi(x+y)-\pi(x)\leq\pi(y)$ contradict each other. The explicit formulae are given once in their classical form, and once in the very general form due to Weil. As applications the error term of the prime number theorem under the Riemann hypothesis and the prime number theorem for almost all short intervals under this assumption are given. Selberg’s sieve is introduced and used to prove the Brun-Titchmarsh inequality, Brun’s upper bound for the number of prime twins, and Romanoff’s theorem stating that a positive proportion of all integers can be written as the sum of a prime number and a power of two. The vertical distribution of the zeros of $\zeta$ is described both unconditional and under the Riemann hypothesis, and it is shown that there are infinitely many zeros on the line $\Re\;s=1/2$. These estimates are used to prove Littlewood’s proof of $\Psi(s)=s+\Omega_{\pm}(\sqrt{x}\log\log\log x)$. Each section is followed by an extensive set of exercises, and each chapter contains a large section of literature references, historical remarks and short descriptions of further results. Means from real analysis used are described in detail in the appendices. The text is very well written and accessible to students. On many occasions the authors explicitly describe basic methods known to everyone working in the field, but too often skipped in textbooks. This book may well become the standard introduction to analytic number theory.