This long-anticipated work shares the aims of its celebrated companion: namely, to provide an introduction for students and a reference for researchers to the techniques, results, and terminology of multiplicative number theory. This volume builds on the earlier one (which served as an introduction to basic, classical results) and focuses on sieve methods. This area has witnessed a number of major advances in recent years, e.g. gaps between primes, large values of Dirichlet polynomials and zero density estimates, all of which feature here. Despite the fact that the book can serve as an entry to contemporary mathematics, it remains largely self-contained, with appendices containing background or material more advanced than undergraduate mathematics. Again, exercises, of which there is a profusion, illustrate the theory or indicate ways in which it can be developed. Each chapter ends with a thorough set of references, which will be essential for all analytic number theorists.

‘This very readable volume, by two great expositors, will provide the student with a thorough grounding in the applications of sieves in multiplicative number theory. The exercises are a real bonus, encouraging the reader to explore in many further directions.’

Roger Heath-Brown OBE FRS - University of Oxford

‘The new text of Montgomery and Vaughan is exquisite! The chapters start off at a gentle pace and accelerate to advanced material. The exercises guide the beginner and expert alike to research topics. The illustrations are inspiring. This book could be used in any intermediate course on analytic number theory and as a reference should be in every mathematician's library.’

Brian Conrey - American Institute of Mathematics

‘Like Volume I, this will be a go-to resource for those wanting a crisp, careful treatment of core topics in analytic number theory, written by two senior figures in the field.’

Ben Green - University of Oxford

‘This excellent book is a must-have for anyone serious about the distribution of primes. The focus is on the various sieve methods, and their application to questions on primes, including the recent developments on bounded gaps between primes. There are numerous exercises in each section, all very well prepared and set up to introduce the reader to the research literature. Indeed, the exercise sections add much value to the book. A true masterpiece!’

Jörg Brüdern, - Georg-August-Universität Göttingen