Number theory has two main branches: additive and multiplicative. Additive number theory studies expressing an integer as the sum of integers in a set; two classical problems in this area are the Goldbach conjecture (about writing even numbers as sums of two primes) and Waring's problem (about writing numbers as sums of n-th powers). Multiplicative number theory deals with prime numbers and related topics, such as factorization and divisors; a key result in this area is Prime Number Theorem. The book under review is one of the most important references in the multiplicative number theory, as its title mentions exactly.
The central topic of book is the distribution of primes, mainly in arithmetic progressions. In general, the author tries to follow the matter historically; as he mentions, the story goes back to Dirichlet's memoir of 1837 on the existence of infinitely many primes in arithmetic progressions, which is birth date of the Analytic Number Theory. Then, we have Riemann's full-effect memoir in the path, which suggested the main lines of investigation in the area. Finally, the author gives some important improvements and concludes the book by introducing some important and challenging problems for further studies. Thus, Davenport's book covers most of the important topics in the theory of distribution of primes and leads the reader to serious research topics, such as approximating the least prime in an arithmetic progression.
The book is very well written. It takes an analytic view of the subject; one may start reading it after having a course in elementary number theory and another course in complex variables and functions. Of course, a little knowledge of algebra (for understanding the nature of characters) is required.
This book is useful for graduate students, researchers and for professors. It is a very good text source specially for graduate levels, but even is fruitful for undergraduates. However, all of 29 sections of the book have no exercises, but students and professors can find many self-discovering homeworks hidden in the heart of its text. In the other hand, a large part of analytic number theory deals with multiplicative problems, and consequently most of its problem books contain exercises in multiplicative number theory; a suitable such problem book is Murty’s “Problems in Analytic Number Theory”.
As I see, the path of most investigations in the theory of distribution of primes derived from the body of this book; like a node on the tree of distribution of primes, which connects classical strong root to modern huge branches. The book understands what exactly is important and what is not. In comparision, there is no other book like it except people who have followed it.
The version under review is third edition and has been revised with both great expertise and care by Hugh L. Montgomery. According to the preface, sections 23-29 were completely rewritten.
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.
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The Distribution of the Primes.- Riemann's Memoir.- The Functional Equation of the L Function.- Properties of the Gamma Function.- Integral Functions of Order 1.- The Infinite Products for xi(s) and xi(s,Zero-Free Region for zeta(s).- Zero-Free Regions for L(s, chi).- The Number N(T).- The Number N(T, chi).- The explicit Formula for psi(x).- The Prime Number Theorem.- The Explicit Formula for psi(x,chi).- The Prime Number Theorem for Arithmetic Progressions (I).- Siegel's Theorem.- The Prime Number Theorem for Arithmetic Progressions (II).- The Pólya-Vinogradov Inequality.- Further Prime Number Sums.