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Arithmetical functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties. (English) Zbl 0807.11001
London Mathematical Society Lecture Note Series. 184. Cambridge: Cambridge University Press. xix, 367 p. (1994).
This book treats diverse aspects of arithmetical functions that typically do not fall into the category of analytic or probabilistic number theory.
After an introductory chapter, chapters II–III treat elementary ways to estimate the mean values of arithmetical functions. Highlights: Wirsing’s mean-value theorem for nonnegative multiplicative functions; Daboussi and Delange’s theorem on the vanishing of the Fourier coefficients of multiplicative functions and its consequence that \(f(n)+ \alpha n\) is uniformly distributed for any irrational \(\alpha\) and additive function \(f\); Daboussi’s elementary proof of the prime number theorem; Saffari and Daboussi’s results on the direct multiplicative decompositions of \(\mathbb{N}\).
The bulk of the book, chapters IV–VIII, is devoted to the study of different classes of almost periodic functions. One starts with the classes of periodic functions and even functions (here it means a function with the property that \(f(n)\) depends only on the \(\text{gcd}(m,n)\) for a certain fixed \(m\)), and forms the completion in certain seminorms. Then the inclusion relations between the arising classes are described, the additive and multiplicative functions in these classes are characterized, Ramanujan expansions and limiting distributions are studied. From the various interesting results we quote only one: a power series \(\sum f(n) x^ n\) formed with an arithmetical function \(f\in {\mathcal B}^ 2\) is non-continuable beyond the unit circle if infinitely many Ramanujan coefficients of \(f\) do not vanish.
Chapter IX presents two important mean value results for multiplicative functions; Wirsing’s theorem with Hildebrand’s elementary proof, and Halász’ theorem with Daboussi and Indlekofer’s proof.
The book comes complete with exercises, graphs of functions (which I found a welcome addition to the text), and photos of mathematicians.
This monograph presents certain aspects of arithmetical functions which, with few exceptions, have not been available in book form. It does not replace the existing works (e.g. P. D. T. A. Elliott’s “Probabilistic number theory”), but it is a very useful addition to them. To understand the book a basic knowledge of number theory and analysis is required; a few less known facts from other branches are given in an appendix.

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions in probabilistic number theory
11K70 Harmonic analysis and almost periodicity in probabilistic number theory