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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 131.04303

**Autor: ** Erdös, Pál; Rényi, Alfréd

**Title: ** On the mean value of nonnegative multiplicative number-theoretical functions (In English)

**Source: ** Mich. Math. J. 12, 321-338 (1965).

**Review: ** Let g(n) be a nonnegative and strongly multiplicative function [i.e. g(mn) = g(m) g(n) for (m,n) = 1 and g(p^{k}) = g(p) for prime p and k = 1,2,...], and let M(g) = **lim**_{N ––> oo} ^{1}/_{N} **sum**_{n \leq N} g(n), if the limit exists. The authors consider the following conditions: (i) the series **sum**_{p} {g(p)-1 \over p} is convergent, (ii) the series **sum**_{p} {[g(p)]^{2} \over p^{2}} is convergent, (iii) for every positive \epsilon, **sum**_{n \leq p \leq N (1+\epsilon)} {g(p) log p \over p} \geq \delta (\epsilon) for N \geq N (\epsilon) with suitable \delta (\epsilon > 0) and N(\epsilon), and prove (Theorem 2) that (i), (ii) and (iii) imply M(g) = **prod**_{p} **[**1+{g(p)-1 \over p} **]**. If (i) and (iii) are satisfied, but (ii) is not, then M(g) exists and is equal to zero (Theorem 6). A similar result is then deduced for general multiplicative functions, and finally some counterexamples are given.

**Reviewer: ** W.Narkiewicz

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

**Index Words: ** number theory

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