Number theory includes some main tools and methods, and then it is all about trying to solve various problems by those methods. Some books focus on the results, and end up repeating a method of proof on a number of similar theorems. The book under review focuses instead on main tools and general methods, and then provides exercises and applications to fix the details in the mind.
Tenenbaum's book is about analytic and probabilistic number theory. It is written seriously and starts off quickly, studying elementary but important matters in the first part. In the second part we find useful complex-analytic methods; as a natural example, the author studies the distribution of primes. The focus on methods makes this second part more useful than the discussion of the distribution of primes in other books.
The third and final part is an introduction to probabilistic methods, mainly studying arithmetic functions. I am not aware of any other book at this level that covers this material. This is one of the things that make this book valuable and important.
The structure of chapters is this: the expert author first describes the methods and then applies it to some important and frequently occurring problems. Then he gives more examples in many well-chosen exercises, mainly selected from some important papers. Each chapter contains some notes related to the topic of chapter; these notes both are historical and topical, and help the interested reader to follow the topics in the references.
The book is very useful for undergraduate and graduate students. It contains clear and well written text, and enough exercises. It is also useful for researchers, because it covers some fundamental methods, and of course for the useful notes. So I can recommend this book for students, researchers and professors, for studying and teaching.
This book originally is written in French; the copy under review is an English translation by C. B. Thomas. The French version has a supplementary volume giving the full solution of all exercises, prepared by the author with collaboration of Jie Wu; this does not seem to have been translated.
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.
Foreword; Notation; Part I. Elementary Methods: Some tools from real analysis; 1. Prime numbers; 2. Arithmetic functions; 3. Average orders; 4. Sieve methods; 5. Extremal orders; 6. The method of van der Corput; Part II. Methods of Complex Analysis: 1. Generating functions: Dirichlet series; 2. Summation formulae; 3. The Riemann zeta function; 4. The Prime Number Theorem and the Riemann Hypothesis; 5. The Selberg–Delange method; 6. Two arithmetic applications; 7. Tauberian theorems; 8. Prime numbers in arithmetic progressions; Part III. Probabilistic Methods: 1. Densities; 2. Limiting distribution of arithmetic functions; 3. Normal order; 4. Distribution of additive functions and mean values of multiplicative functions; 5. Integers free of large prime factors. The saddle-point method; 6. Integers free of small prime factors; Bibliography; Index.