Analytic number theory, originated by Euler and developed by Dirichlet and Riemann, studies properties of numbers by utilizing tools from analysis. Probabilistic number theory is much younger, from about one hundred years ago. It studies problems of a probabilistic nature related to the properties of numbers, mainly using tools from analytic number theory. Thus, a book in analytic and probabilistic number theory seems very natural and necessary. The book under review is the third edition of Tenenbaum's impressive work in this area.
The book has a classical flavor though it considers very recent improvements and covers a wide range of important topics in both analytic and probabilistic number theory. It includes, for example, the study of the distribution of prime numbers both by elementary methods and by using the properties of the Riemann zeta function. It also discusses arithmetic functions and their average and extremal orders. Both of these are very classical subjects.
The book is arranged in three parts entitled “elementary,” “complex analysis,” and “probabilistic” methods. Each chapter ends with notes giving a historical picture of the chapter's material and also mentioning developments and improvements. The book contains 307 exercises, distributed in the various chapters but numbered continuously in whole book. These exercises are very good. Indeed, many of them are serious results from papers, which will open some very good research paths for readers.
In comparison with the previous English edition, which was a translation of the first edition in French, several chapters of the book have grown. This includes the chapters on sieve methods, the prime number theorem and the Riemann hypothesis, Tauberian theorems, primes in arithmetic progressions, integers free of large prime factors and the saddle-point method, and integers free of small prime factors. A chapter on Diophantine approximation, and a very interesting chapter on the Euler Gamma function and its analytic properties are new to this edition.
The main point about the book is that its author is an eminent expert in analytic and probabilistic number theory and has written a remarkable number of papers and books. I believe that this book is a very good source for graduate students studying analytic and probabilistic number theory. The readers will find both the origins of the subject and serious research paths in this book. I strongly recommend this book for students, instructors, and researchers in analytic and probabilistic number theory. Also, considering the analysis used in the book, it seems that this book is also a good source for people working on elementary and complex analysis who are seeking some meaningful applications in other branches of mathematics.
Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.