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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. Transl. from the Russsian by G.A. Kandall. Ed. by Ben Silver. (English)
Translation 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\sb 1+\dots +x\quad s\sb n,\quad n\to \infty,\quad n Reviewer: Wolfgang Schwarz (Frankfurt / Main)
Keywords: Wintner’s theorem; Wirsing’s theorem; Axer’s theorem; Novoselov’s theory; distribution of values of arithmetical functions; Bredikhin theorem on counting functions of arithmetical semigroups; asymptotic density; prime independent multiplicative functions; Tauberian theorem of Hardy-Littlewood; Ingham Tauberian theorem for partitions; Esseen’s inequality; distribution functions; additive problems; increasing number of summands; lattice points in parallelepipeds; local limit theorem; Freiman’s theorem; asymptotic formula; number of solutions; Hardy-Ramanujan partition formula; arithmetical functions; logarithmic density; mean-value theorems; almost-periodic functions; Polyadic analysis; Turán-Kubilius inequality; multiplicative functions; Erdős-Wintner theorem; Delange’s theorem; Euler function; Erdős-Kac theorem; strongly additive functions; asymptotic expansions; asymptotic formulas; bibliography
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