Analytic Number Theory is an old subject, going back to Euler. Since then, and especially after Dirichlet and Riemann, a large number of great mathematicians have contributed to it. A search on “Analytic Number Theory” finds a remarkable number of classical and modern books with this title, which cover many different topics. It is clear that creating and writing another book in this area is not an easy task. The expert authors of the book under review, who are well-known for their many high quality contributions, have done a good job with this task as well.
The book retains the classical skeleton of the subject, covering such topics as the proof of prime number theorem (Newman’s proof, but with more details), sieve methods, and prime numbers in arithmetic progression. On the other hand, in all parts of the book we see that the authors try to guide their readers — graduate students and professors, probably — toward recent research in the area. Indeed, the book is full of research ideas and highlights important directions of thought that are essential for doing research in the subject.
Chapters 13–15 treat the standard topic of “prime numbers in arithmetic progression” and “Dirichlet’s Theorem”. The authors give a very clear account of preliminary matters, including Dirichlet characters and Dirichlet L-functions and give the classical proof of Dirichlet’s theorem for prime numbers in arithmetic progressions. This is what may be found in many similar books, for example those of Davenport, Tenenbaum, and Montgomery and Vaughan. But there are more theorems about primes in arithmetic progression, some of which have important applications. For example, the Bombieri-Vinogradov Theorem is used to study distribution functions of arithmetical functions running over shifted primes. What is different (and nice) about the book under review is that in chapter 15 it describes selected applications of primes in arithmetic progressions, including that one.
The authors keep their focus on current research, for example in Chapters 6 and 7, where they study “Global and Local Behavior of Arithmetic Functions”, in Chapter 9 which is about “Smooth Numbers”, and in Chapter 11 which studies “The abc Conjecture and Some of Its Applications”. Similarly, in Chapter 16 they discuss recently developed subject of “The Index of Composition of an Integer”; this seems to be the first treatment of the subject in a textbook. These features also help justify the subtitle of the book: “Exploring the Anatomy of Integers.”
The book is very well-written, introducing topics in a fluent and readable way. The authors state and use a number of important theorems, such as Bombieri-Vinogradov Theorem mentioned above, without proofs. This results in a friendly volume that allows graduate students to get to important topics very fast. Thus, the book is very suitable as a text for graduate courses. Moreover, all chapters end with very good exercises, and fully-detailed solutions of even-numbered problems are included (a remarkable feature at this level).
I strongly recommend this book for graduate students who are hoping to start a research project in analytic number theory and for professors who are going to teach and introduce their students in the path of research. Both groups will find the book accessible and friendly.
Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.