Distribution of prime numbers

Prime numbers have fascinated mathematicians since the time of Euclid. This book presents some of our best tools to capture the properties of these fundamental objects, beginning with the most basic notions of asymptotic estimates and arriving at the forefront of mathematical research. Detailed proofs of the recent spectacular advances on small and large gaps between primes are made accessible for the first time in textbook form. Some other highlights include an introduction to probabilistic methods, a detailed study of sieves, and elements of the theory of pretentious multiplicative functions leading to a proof of Linnik's theorem. Throughout, the emphasis has been placed on explaining the main ideas rather than the most general results available. As a result, several methods are presented in terms of concrete examples that simplify technical details, and theorems are stated in a form that facilitates the understanding of their proof at the cost of sacrificing some generality. Each chapter concludes with numerous exercises of various levels of difficulty aimed to exemplify the material, as well as to expose the readers to more advanced topics and point them to further reading sources.

It's clear that Koukoulopoulos had a marvelous time putting together this beautiful material and producing a very readable and pedagogically sound text (replete with good exercises). The book is well-paced and reads very well. The careful reader, with pencil and paper in hand, keen to do exercises galore and have fun doing so, will learn a lot of beautiful number theory and find out marvels about the secret life of the set of primes: they are elusive but not unyielding. --Michael Berg, Loyola Marymount University

...this is an excellent book introducing the reader to a wealth of modern techniques for studying prime numbers. There is a lot of new material here that has never appeared before in book form. The author took great care in explaining both the intuition behind this very technical subject and in providing the 'best' proofs, especially proofs that are short and understandable. The book will be an excellent introduction to anybody interested in primes at a research level (or rather, interested in quickly reaching this level). --Maksym Radziwi, University of Texas at Austin

The book under review is a really beautiful guide to the mysteries involving the distribution of prime numbers. The book is written in such a manner to introduce beginning graduate students as well as advanced undergraduate students to the related methods of analytic number theory. A very nice aspect of this work is that the author gives emphasis on demonstrating the main ideas involved, thus making the presentation and flow of the book much more natural and reader friendly. --Michael Th.Rassias (Zürich)

Includes list of notations, appendices, bibliography and index.