The sieve method has undergone a rapid development in the last few decades and has become an important tool in analytic and combinatorial number theory. This is also reflected in the fact that several very good books devoted to this topic appeared in the last newsletter. This evokes an impression that all the important facets of development in the surrounding theory have already been covered in monograph form. But sometimes this is not the most important aspect. The approach used to describe the main ingredients of the idea makes the point. All this can be found in the book.

The author writes in the preface: “This book charts my own mathematical journey and the adventures of the others in using sieve methods to generate primes”. These words illustrate the contents of the book. The author describes very carefully the main ingredients and motivations behind the development of ideas and their impact on solutions of presented problems. The prime-detecting aspects of modern sieve methods are demonstrated in the way they were used to handle such important themes as primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes (including the recent work of Friedlander and Iwaniec on primes that are a sum of a square and a fourth power) and Heath-Brown’s work on primes represented as a cube plus twice a cube. The book is written in a very accessible style for a wide spectrum of readers from graduate students to researchers wishing to learn deep ideas of modern sieve methods. Besides mathematical ideas, the presentation also contains many important historical comments, which make the book useful for a general mathematical audience trying to orient themselves in the evolution of the main techniques applied in sieve methods.