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Introduction to analytic and probabilistic number theory. Transl. from the 2nd French ed. by C.B.Thomas. (English) Zbl 0831.11001
Cambridge Studies in Advanced Mathematics. 46. Cambridge: Cambridge Univ. Press. xiv, 448 p. £45.00; \$ 64.95 (1995).

[For a review of the French original (Univ. Nancy, 1990) see Zbl 0788.11001.]

This well-written monograph arose from graduate courses given in Bordeaux, Paris and Nancy; it contains an enormous wealth of material from number theory, many [difficult] exercises, requiring considerable skilfulness, and many references in the bibliography. Informative notes to every section give hints to the bibliography, to the history of the investigations under consideration and to more general or more recent results. The author has “been guided by the constant concern of emphasizing the methods rather more than the results, a strategy which we believe to be specifically heuristic. ... Without aiming at complete originality, the text tries to avoid well-trodden paths.”

The book is divided into three parts: I. Elementary methods, II. Methods of complex analysis, III. Probabilistic methods.

The author treats in Chapter I: the elementary theory of prime numbers (including Chebyshev’s theorem and Mertens’ formula), arithmetical functions (average results, extremal orders), Brun’s sieve method and the “large sieve”, and the estimation of exponential sums using van der Corput’s method (including an application to Voronoi’s theorem).

Chapter II presents the theory of Dirichlet series, including Perron’s formula, bounds for ${\zeta }^{\text{'}}/\zeta$, $1/\zeta$, $log\zeta$, the functional equation, and zero-free regions of the Riemann zeta function $\zeta \left(s\right)$. The prime number theorem (with remainder term $x·exp\left\{-c\sqrt{logx}\right\}\right)$ is proved, the Selberg-Delange method of obtaining an asymptotic expansion for ${\sum }_{n\le x}{a}_{n}$, where $F\left(s\right)={\sum }_{n=1}^{\infty }{a}_{n}{n}^{-s}$ is a Dirichlet series behaving similarly to ${\left(\zeta \left(s\right)\right)}^{z}$, is applied to study integers having exactly $k$ prime factors, and to study the average distribution of the divisors of integers. The Tauberian theorem of Hardy-Littlewood- Karamata, including a version with remainder term, and an “effective” form of the Ikehara theorem are proved. Finally, the prime number theorem in arithmetic progressions is proved, and the Siegel-Walfisz theorem is stated. The Bombieri-Vinogradov prime number theorem is given in the notes.

Chapter III, Probabilistic methods, introduces different concepts of density, and relates limit distributions of arithmetical functions to characteristic functions by Lévy’s continuity theorem. The Turán- Kubilius inequality (and its dual) is proved and applied to deduce the Hardy-Ramanujan theorem on large deviations of $|\omega \left(n\right)-loglogN|$. Effective upper estimates of ${\sum }_{n\le x}f\left(n\right)$ are given for non-negative multiplicative functions $f$. For real-valued additive functions the Erdös-Wintner theorem and the Erdös-Kac theorem, concerning the weak convergence of the distribution function

$\frac{1}{N}·#\left\{n\le N;\phantom{\rule{4pt}{0ex}}\omega \left(n\right)\le loglogN+y\sqrt{loglogN}\right\},$

are proved. For multiplicative functions, mean- value theorems of Delange, Wirsing and Halász are proved.

Denoting by ${P}^{+}\left(n\right)$ the maximal prime divisor of $n$, the function

${\Psi }\left(x,y\right)=#\left\{n\le x;\phantom{\rule{4pt}{0ex}}{P}^{+}\left(n\right)\le y\right\}$

is studied thoroughly; estimates and asymptotic formulae due to Rankin, de Bruijn, Ennola, Alladi, Saias, Hildebrand are proved. Similarly, in the last section, the function

${\Phi }\left(x,y\right)=#\left\{n\le x;\phantom{\rule{4pt}{0ex}}{P}^{-}\left(n\right)>y\right\}$

is studied, where ${P}^{-}\left(n\right)$ denotes the least prime factor of $n$.

Some of the results given are new or unpublished in book form, for example, “the results derived from the Selberg-Delange method ..., ..., and the study of the function ${\Phi }\left(x,y\right)$ via the saddle-point method.”

The reviewer thinks this monograph is a most welcome addition to the literature on analytic and probabilistic number theory.

MSC:
 11-01 Textbooks (number theory) 11-02 Research monographs (number theory) 11Nxx Multiplicative number theory 11Kxx Probabilistic number theory 11Mxx Analytic theory of zeta and $L$-functions 11Lxx Exponential sums; character sums 11N35 Sieves 11N37 Asymptotic results on arithmetic functions 11N56 Rate of growth of arithmetic functions 11A41 Elementary prime number theory 11A25 Arithmetic functions, etc. 11K65 Arithmetic functions (probabilistic number theory) 11L40 Estimates on character sums 11N60 Distribution functions (additive and positive multiplicative functions) 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11M45 Tauberian theorems 11N05 Distribution of primes 11N13 Primes in progressions 11L07 Estimates on exponential sums