Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 022.00903
Autor: Erdös, Paul; Wintner, Aurel
Title: Additive arithmetical functions and statistical independence. (In English)
Source: Amer. J. Math. 61, 713-721 (1939).
Review: Important results are obtained concerning additive functions, i.e. functions f(n) which satisfy f(n1 n2) = f(n1)+f(n2), whenever (n1,n2) = 1; so that f(n) is determined by the values of f(pk), for all primes p and all k. It is shown that such a function has an asymptotic distribution function \sigma if and only if sum p-1 g(p) and sum' p-1 g(p)2 are convergent, when g(p) = f(p) or g(p) = 1 according as |f(p)| < 1 or |f(p)| \geq 1. Furthermore, if \sigmap is the asymptotic distribution function of the function fp(n), which is defined by fp(n) = f(pk) if pk|n and pk+1 \nmid n, then \sigma is the infinite convolution of the \sigmap and the above condition for the existence of \sigma is identical with the condition that this infinite convolution be convergent. The complete proof of which large parts are given in earlier publications [cf. P. Erdös, J. London Math. Soc. 13, 119-127 (1938; Zbl 018.29301)] is long and involves delicate operations with prime numbers related to Brunn's method.
Reviewer: E.R.van Kampen (Baltimore)
Classif.: * 11N60 Distribution functions (additive and positive multipl. functions)
11K65 Arithmetic functions (probabilistic number theory)
Index Words: Number theory
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