I would like to start my review of this book by quoting from the preface to the first edition: "In order to become proficient in mathematics, or in any subject," writes André Weil, "the student must realize that most topics involve only a small number of basic ideas." After learning these basic concepts and theorems, the student needs "drill in routine exercises, by which the necessary reflexes in handling such concepts may be acquired… There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems." Weil's insightful observation becomes especially important at the graduate and research level. This is the viewpoint of the book under review.
The second edition of the book has eleven chapters (one more than its first edition), mainly covering the analytic theory of the distribution of the prime numbers and the main tools of this area: primes in arithmetic progressions, the prime number theorem, the Selberg class of functions, sieve methods, p-adic methods, and (new in the present edition) the important topic of equidistribution. Each subject gets its own chapter covering their required fundamental topics, all of which are well motivated. All chapters contain some landmark theorems, which open up a method of attack for many of the exercises that follow. Thus, the book can be used both as a problem book (as its title shows) and also as a textbook (as the series in which the book is published shows). Finally, since the exercises state many of the main results in the topics under discussion, the first part (problems) of the book can be considered a brief handbook on the theory of distribution of primes.
The book is ideal as a text for a first course in analytic number theory, either at the senior undergraduate or the graduate level. Also, it can be used as a supplementary book beside other important sources of the area, such as Davenport's Multiplicative Number Theory .
I believe that this book will be very useful for students, researchers and professors. It is well written and well typeset; the few little typos don't diminish the beauty of the book.
Mehdi Hassani is a "co-tutelle" Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.