##
**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. (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 \(\zeta'/ \zeta\), \(1/ \zeta\), \(\log \zeta\), the functional equation, and zero-free regions of the Riemann zeta function \(\zeta (s)\). The prime number theorem (with remainder term \(x\cdot \exp\{-c \sqrt {\log x}\})\) is proved, the Selberg-Delange method of obtaining an asymptotic expansion for \(\sum_{n\leq x} a_n\), where \(F(s)= \sum_{n=1}^\infty a_n n^{-s}\) is a Dirichlet series behaving similarly to \((\zeta (s))^z\), 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 \(|\omega (n)-\log \log N|\). Effective upper estimates of \(\sum_{n\leq x} f(n)\) 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 \[ {\textstyle {1\over N}}\cdot \#\left\{n\leq N;\;\omega(n)\leq \log \log N+ y\sqrt {\log \log N}\right\}, \] are proved. For multiplicative functions, mean-value theorems of Delange, Wirsing and Halász are proved.

Denoting by \(P^+ (n)\) the maximal prime divisor of \(n\), the function \[ \Psi (x, y)= \#\{n\leq x;\;P^+ (n)\leq y\} \] 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 \[ \Phi (x, y)= \#\{n\leq x;\;P^- (n)> y\} \] is studied, where \(P^- (n)\) 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 \(\Phi (x, y)\) via the saddle-point method.”

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

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 \(\zeta'/ \zeta\), \(1/ \zeta\), \(\log \zeta\), the functional equation, and zero-free regions of the Riemann zeta function \(\zeta (s)\). The prime number theorem (with remainder term \(x\cdot \exp\{-c \sqrt {\log x}\})\) is proved, the Selberg-Delange method of obtaining an asymptotic expansion for \(\sum_{n\leq x} a_n\), where \(F(s)= \sum_{n=1}^\infty a_n n^{-s}\) is a Dirichlet series behaving similarly to \((\zeta (s))^z\), 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 \(|\omega (n)-\log \log N|\). Effective upper estimates of \(\sum_{n\leq x} f(n)\) 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 \[ {\textstyle {1\over N}}\cdot \#\left\{n\leq N;\;\omega(n)\leq \log \log N+ y\sqrt {\log \log N}\right\}, \] are proved. For multiplicative functions, mean-value theorems of Delange, Wirsing and Halász are proved.

Denoting by \(P^+ (n)\) the maximal prime divisor of \(n\), the function \[ \Psi (x, y)= \#\{n\leq x;\;P^+ (n)\leq y\} \] 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 \[ \Phi (x, y)= \#\{n\leq x;\;P^- (n)> y\} \] is studied, where \(P^- (n)\) 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 \(\Phi (x, y)\) via the saddle-point method.”

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

Reviewer: Wolfgang Schwarz (Frankfurt am Main)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Nxx | Multiplicative number theory |

11Kxx | Probabilistic theory: distribution modulo \(1\); metric theory of algorithms |

11Mxx | Zeta and \(L\)-functions: analytic theory |

11Lxx | Exponential sums and character sums |

11N35 | Sieves |

11N37 | Asymptotic results on arithmetic functions |

11N56 | Rate of growth of arithmetic functions |

11A41 | Primes |

11A25 | Arithmetic functions; related numbers; inversion formulas |

11K65 | Arithmetic functions in probabilistic number theory |

11L40 | Estimates on character sums |

11N60 | Distribution functions associated with additive and positive multiplicative functions |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11M45 | Tauberian theorems |

11N05 | Distribution of primes |

11N13 | Primes in congruence classes |

11L07 | Estimates on exponential sums |

### Keywords:

\(L\)-series; densities; Delange’s theorem; Wirsing’s theorem; Halász’ theorem; integers without large prime factors; integers free of small prime factors; large sieve; analytic number theory; exercises; prime numbers; arithmetical functions; extremal orders; Brun’s sieve; estimation of exponential sums; van der Corput’s method; Dirichlet series; Perron’s formula; functional equation; zero-free regions of the Riemann zeta function; prime number theorem; Selberg-Delange method; integers having exactly \(k\) prime factors; Tauberian theorem; Ikehara theorem; Siegel-Walfisz theorem; Bombieri-Vinogradov prime number theorem; limit distributions of arithmetical functions; Turán-Kubilius inequality; Erdös-Wintner theorem; Erdös-Kac theorem; mean-value theorems; saddle-point method; probabilistic number theory### Citations:

Zbl 0788.11001
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\textit{G. Tenenbaum}, Introduction to analytic and probabilistic number theory. Transl. from the 2nd French ed. by C. B.Thomas. Cambridge: Cambridge Univ. Press (1995; Zbl 0831.11001)

### Online Encyclopedia of Integer Sequences:

The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

a(n) = Sum_{k=1..n} 2^bigomega(k).

Number of numbers of the form p*q (p, q distinct primes) less than or equal to n.

Number of partitions of n with a product <=n.

Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2)))