The distribution of prime numbers. (English) Zbl 1468.11001

Koukoulopoulos, Dimitris

Graduate Studies in Mathematics 203. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4754-0/hbk; 978-1-4704-5420-3/ebook). xii, 356 p. (2019).

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.
The book is composed by 30 chapters which have been partitioned accordingly to 6 parts. The first part presents some basic useful principles. The second part progresses to the presentation of methods of complex analysis and harmonic analysis used in the topics treated. The third part is devoted to multiplicative functions and the anatomy of integers. The following two parts investigate sieve methods and bilinear forms and the last part is devoted to the investigation of local aspects of the distribution of prime numbers. Each chapter is accompanied by a list of interesting exercises, which are carefully selected to examine the concepts studied as well as to guide the reader to delve into the discovery of more advanced topics. Finally, the book also includes three useful appendices.
Apart from the didactic aspect of this book, it could also serve as a very useful references source for a plethora of beautiful results in the topics treated.
Overall, the book – which is devoted to a very classical and active domain of mathematics – is very well written and it certainly belongs to all libraries of universities and research institutes with a Mathematics Department.

Reviewer: Michael Th. Rassias (Zürich)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11N05	Distribution of primes
11Mxx	Zeta and \(L\)-functions: analytic theory

Keywords:

prime numbers; distribution; asymptotics; sieve methods; bilinear methods; multiplicative functions