Introduction to analytic and probabilistic number theory. (Introduction à la théorie analytique et probabiliste des nombres.) (French) Zbl 0788.11001

Tenenbaum, G.

Institut Élie Cartan, Université de Nancy I. 13. Nancy: Université de Nancy. xii, 499 p. (1990).

This is a very nicely written book that any graduate student working in analytic or probabilistic number theory, or indeed any mathematician wishing to gain some new insight into the topics presented, should find both helpful and stimulating. The book falls into three parts: Elementary methods, Complex analytic methods, Probabilistic methods.
The first part contains an exposition of standard elementary results concerning prime numbers and familiar arithmetic functions. It includes Chebyshev’s inequalities for \(\pi(x)\), the maximal order and average order of magnitude of arithmetic functions such as the divisor functions, Euler’s function, and the additive functions \(\omega(n)\) and \(\Omega(n)\). The chapter on sieve methods ends by establishing a form of the Brun- Titchmarsh theorem on the number of primes in an arithmetic progression that lie in a short interval.
The second part concentrates on the analytic approach to questions in prime number theory and related topics. Properties of Dirichlet series and the Riemann zeta-function are established, and there is a chapter on Tauberian theorems. Proofs are given of the prime number theorem, with a good error term, and a corresponding uniform result for primes in arithmetic progression. Asymptotic expansions for the number of positive integers \(n\leq x\) with exactly \(k\) prime factors, counted with and without multiplicity and uniform for \(k\ll\log\log x\), are established using a method due to Selberg and Delange.
After some preliminaries concerning the concept of density and the distribution function of an arithmetic function, the third part goes on to describe standard results concerned with the normal order of certain arithmetic functions, the limiting distribution of additive functions and the mean value of multiplicative functions. In the last two chapters, the function \(\Psi(x,y)\), \(\Phi(x,y)\), defined to be the number of positive integers \(n\leq x\) with no prime factor \(p>y\), every prime factor \(p>y\), respectively, are studied; the results described range from the classical to the very recent.
At the end of each chapter, there is a section giving historical notes and stating related and sometimes relatively new results not established in the text. There are also some challenging exercises to test the reader. An extensive and very useful bibliography is provided at the end of the book.
It will be of great value to the interested reader to have such a variety of topics collected together and expounded in one book. As one would expect, there is an overlap between material in the first two parts in particular and the contents of various standard textbooks. However the selection and presentation of the topics discussed here reflect the author’s own interests and personal approach as well as bringing the reader up to the forefront of recent research work. The exposition is clear and given in sufficient detail for the hardworking novice to follow it. This is altogether a worthy and useful addition to the collection of books on analytic number theory.

Reviewer: E.J.Scourfield (Egham)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11N37	Asymptotic results on arithmetic functions
11K65	Arithmetic functions in probabilistic number theory
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11A25	Arithmetic functions; related numbers; inversion formulas
11Nxx	Multiplicative number theory
11Mxx	Zeta and \(L\)-functions: analytic theory

Keywords:

analytic number theory; elementary methods; complex analytic methods; probabilistic methods; primes in arithmetic progression; integers without large prime factors; probabilistic number theory; arithmetic functions; sieve methods; Brun-Titchmarsh theorem; Dirichlet series; Riemann zeta- function; Tauberian theorems; prime number theorem; distribution function; normal order; limiting distribution; additive functions; mean value; multiplicative functions; bibliography