Lectures on sieve methods and prime number theory
This book is not directed at beginners in either sieve methods or in prime number theory; but the reader with some knowledge of these topics will value it not only for the proofs and arguments contained in the text, but also for the guide to the literature supplied by the notes at the end of each of the six chapters. The aim of the book is to describe the unification of sieve methods with analytic ones in number theory that has been a hallmark of much work in the subject over the last decade or so, and the choice of topics in sieve methods that form the first half of the book has been made with this end in view. Thus the reader will not find more than a passing mention of the conventional treatment of Selberg's sieve method; he will find a discussion of Selberg's sieve for intervals that makes use of the large sieve inequality, followed by a hybridisation of these ideas with those involved in the multiplicative aspects of the applications of this inequality. This is followed by a discussion of the combinatorial sieve method, leading to a treatment of Iwaniec's bilinear form of the error term in the linear sieve. Part II of the text, on topics in prime number theory, opens with a chapter on zero-free regions for Riemann's zeta-function and for Dirichlet's L-functions. For Vinogradov's zero-free region for the Riemann zeta-function the author presents an inference from estimates of exponential sums that dispenses with the usual convexity argument of Borel-Caratheodory type, replacing it with an argument of a sieve- theoretic nature. The Deuring-Heilbronn phenomenon for zeros of L- functions is discussed in the same spirit. After a chapter devoted to estimates for densities of possible zeros of these functions the account reaches its climax with a treatment of questions relating to gaps between consecutive primes and to the least prime in an arithmetic progression.
zero- free regions for Riemann's zeta-function
least prime in an arithmetic progression
scientific article; zbMATH DE number 3849304