London Mathematical Society Monographs Series 33. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12437-7/hbk). xiv, 362 p. $ 65.00; £ 38.95 (2007).

This monograph describes both the theory of the small sieve (principally in the case of sieving dimension one) and its applications to prime numbers. In the early days of the sieve it had been believed that the methods were incapable of proving the existence of primes, because of the “parity phenomenon”, but this idea is long out of date, as the book amply proves.
We start with chapters on the history of sieve ideas, on the Vaughan identity and its generalizations, on the Rosser--Iwaniec sieve, and on the “Alternative Sieve”. This last chapter introduces a very fruitful technique. If one wants to find primes in a difficult set $\mathcal{A}$ one looks for a simpler set $\mathcal{B}$ for which one can calculate the sifting functions which occur. Then, if the sieve data $\#\mathcal{A}_d$ and $\#\mathcal{B}_d$ match up for a suitable set of values of $d$, one will be able to estimate sifting functions for $\mathcal{A}$ in terms of the corresponding functions for $\mathcal{B}$. A simple example is discussed, in which $\mathcal{A}$ is the Piatetski-Shapiro sequence $[n^{\gamma}]$ and $\mathcal{B}$ is the set of integers in an interval $(x,2x]$.
These ideas are then put into practice for a simple upper bound sieve, which is fed into Chebyshev’s method to show that any interval $(x,x+x^{1/2}]$ with $x$ sufficiently large contains an integer whose largest prime factor exceeds $x^{0.74}$.
The subsequent chapters present further applications, developing these ideas. They concern primes in the interval $[x,x+x^{\theta}]$, the Brun--Titchmarsh Theorem on average, and primes in almost-all short intervals. The vector sieve is then introduced, with an application to Goldbach numbers in short intervals.
The last three substantive chapters show how sieve methods can be used in algebraic number fields. The first of these presents results on the distribution of Gaussian primes in narrow sectors and small discs. The second describes the work of Fouvry and Iwaniec on primes $a^2+p^2$, and of Friedlander and Iwaniec on primes $a^2+b^4$. Finally, there is a discussion of the reviewer’s work on primes of the form $a^3+2b^3$. The book ends with an epilogue, five appendices covering basic analytic techniques, and a bibliography of 168 items.
This volume takes the reader right up to the leading edge of current research. The area is one in which multiple case-by-case analyses and detailed calculations are sometimes unavoidable. The book is therefore unsuitable for lazy students. However those who want to make a serious study of the area will appreciate the author’s unified approach to the methods which have been employed, and the wealth of applications described. The book is to be recommended to anyone with an interest in sieves, from beginning PhD students to established researchers.