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**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(n_{1} n_{2}) = f(n_{1})+f(n_{2}), whenever (n_{1},n_{2}) = 1; so that f(n) is determined by the values of f(p^{k}), 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 \sigma_{p} is the asymptotic distribution function of the function f_{p}(n), which is defined by f_{p}(n) = f(p^{k}) if p^{k}|n and p^{k+1} \nmid n, then \sigma is the infinite convolution of the \sigma_{p} 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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