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**Introduction to analytic number theory. Transl. from the Russian by G. A. Kandall. Ed. by Ben Silver. Appendix by P. D. T. A. Elliott.**
*(English)*
Zbl 0641.10001

Translations of Mathematical Monographs, 68. Providence, RI: American Mathematical Society (AMS). vi, 320 p.; $ 114.00 (1988).

[For a review of the Russian original (Nauka, Moskva 1971) see Zbl 0231.10001.]

This monograph contains several topics from the analytic theory of numbers, which generally are not treated in textbooks on analytic number theory.

The first chapter presents background from analysis: the Tauberian theorem of Hardy-Littlewood (also in G. Freud’s version with remainder term) and the Ingham Tauberian theorem for partitions are proved, and Esséen’s inequality, estimating the difference of two distribution functions in terms of their characteristic functions, is given.

The second chapter deals with additive problems with an increasing number of summands. Th. Schneider’s lemma on lattice points in parallelepipeds is proved in a probabilistic setting, applications of the local limit theorem of probability theory to number theory are sketched, and Freiman’s theorem concerning an asymptotic formula for the number of solutions of \(N=x\quad s_ 1+\dots +x\quad s_ n,\quad n\to \infty,\quad n<C\cdot N,\) is proved. Connections between the counting functions \(\pi_ G(x)\) and \(\nu_ G(x)\) in arithmetical semigroups \(G\) (due to Bredikhin) are given. Finally the Hardy-Ramanujan partition formula and Ingham’s general results on partitions are proved.

The third chapter deals with arithmetical functions; first the concepts of asymptotic and logarithmic density are discussed, the mean-value theorems of Wintner and Axer are given. Polyadically continuous functions and (B1-) almost-periodic functions are defined, and their properties are studied. A local distribution law for integer-valued additive functions is proved. Finally, E. V. Novoselov’s “Polyadic analysis and applications” (which partly was printed in journals not easily accessible) is presented.

Chapter 4 is concerned with further rather deep results on multiplicative functions. At first, a theorem of Drozdova and Freiman on a (sharp) upper bound for the values of nonnegative prime-independent multiplicative functions is given [more precise results are due to E. Heppner, Arch. Math. 24, 63–66 (1973; Zbl 0254.10038)]. Secondly, E. Wirsing’s famous mean-value theorem for multiplicative functions [Math. Ann. 143, 75–102 (1961; Zbl 0104.042)] is proved, following Wirsing’s proof. The Turán-Kubilius inequality is deduced. Next, Delange’s mean- value theorem for multiplicative functions \(f\) with modules \(| f| \leq 1\) is proved; Halász’s theorem is quoted. The Erdős-Wintner theorem is deduced as a corollary to Delange’s theorem. The distribution of the values of the Euler function is treated, and the Erdős-Kac theorem for strongly additive functions is proved. Finally, asymptotic expansions and asymptotic formulas for some special multiplicative functions are proved.

Following the bibliography with 157 items, an informative appendix written by P. D. T. A. Elliott is given, with additional hints and references to more recent research papers.

Postnikov’s “Introduction to analytic number theory” certainly is not an “Introduction”; it “represents an interesting personal selection of a number of topics from analytic number theory”, avoiding those topics already sufficiently covered in the literature. Therefore this monograph seems to be an important supplement to existing textbooks on analytic number theory, recommendable to researchers, teachers and graduate students, interested in analytic number theory.

This monograph contains several topics from the analytic theory of numbers, which generally are not treated in textbooks on analytic number theory.

The first chapter presents background from analysis: the Tauberian theorem of Hardy-Littlewood (also in G. Freud’s version with remainder term) and the Ingham Tauberian theorem for partitions are proved, and Esséen’s inequality, estimating the difference of two distribution functions in terms of their characteristic functions, is given.

The second chapter deals with additive problems with an increasing number of summands. Th. Schneider’s lemma on lattice points in parallelepipeds is proved in a probabilistic setting, applications of the local limit theorem of probability theory to number theory are sketched, and Freiman’s theorem concerning an asymptotic formula for the number of solutions of \(N=x\quad s_ 1+\dots +x\quad s_ n,\quad n\to \infty,\quad n<C\cdot N,\) is proved. Connections between the counting functions \(\pi_ G(x)\) and \(\nu_ G(x)\) in arithmetical semigroups \(G\) (due to Bredikhin) are given. Finally the Hardy-Ramanujan partition formula and Ingham’s general results on partitions are proved.

The third chapter deals with arithmetical functions; first the concepts of asymptotic and logarithmic density are discussed, the mean-value theorems of Wintner and Axer are given. Polyadically continuous functions and (B1-) almost-periodic functions are defined, and their properties are studied. A local distribution law for integer-valued additive functions is proved. Finally, E. V. Novoselov’s “Polyadic analysis and applications” (which partly was printed in journals not easily accessible) is presented.

Chapter 4 is concerned with further rather deep results on multiplicative functions. At first, a theorem of Drozdova and Freiman on a (sharp) upper bound for the values of nonnegative prime-independent multiplicative functions is given [more precise results are due to E. Heppner, Arch. Math. 24, 63–66 (1973; Zbl 0254.10038)]. Secondly, E. Wirsing’s famous mean-value theorem for multiplicative functions [Math. Ann. 143, 75–102 (1961; Zbl 0104.042)] is proved, following Wirsing’s proof. The Turán-Kubilius inequality is deduced. Next, Delange’s mean- value theorem for multiplicative functions \(f\) with modules \(| f| \leq 1\) is proved; Halász’s theorem is quoted. The Erdős-Wintner theorem is deduced as a corollary to Delange’s theorem. The distribution of the values of the Euler function is treated, and the Erdős-Kac theorem for strongly additive functions is proved. Finally, asymptotic expansions and asymptotic formulas for some special multiplicative functions are proved.

Following the bibliography with 157 items, an informative appendix written by P. D. T. A. Elliott is given, with additional hints and references to more recent research papers.

Postnikov’s “Introduction to analytic number theory” certainly is not an “Introduction”; it “represents an interesting personal selection of a number of topics from analytic number theory”, avoiding those topics already sufficiently covered in the literature. Therefore this monograph seems to be an important supplement to existing textbooks on analytic number theory, recommendable to researchers, teachers and graduate students, interested in analytic number theory.

Reviewer: Wolfgang Schwarz (Frankfurt / Main)

### MSC:

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

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

11N37 | Asymptotic results on arithmetic functions |

11K65 | Arithmetic functions in probabilistic number theory |

11P05 | Waring’s problem and variants |

11N05 | Distribution of primes |

11N13 | Primes in congruence classes |

11N80 | Generalized primes and integers |

11N99 | Multiplicative number theory |

11P81 | Elementary theory of partitions |

11D72 | Diophantine equations in many variables |