Multiplicative Number Theory II : Primes and Sieves

Long anticipated, this work focuses on sieve methods and their applications to areas such as the gaps between prime numbers, large values of Dirichlet polynomials, and zero density estimates. With numerous exercises and a thorough set of references, this book will prove essential reading for analytic number theorists.

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.