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(pk) = g(p) for prime p and k = 1,2,...], and let M(g) = limN ––> oo 1/N sumn \leq N g(n), if the limit exists. The authors consider the following conditions: (i) the series sump {g(p)-1 \over p} is convergent, (ii) the series sump {[g(p)]2 \over p2} is convergent, (iii) for every positive \epsilon, sumn \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) = prodp [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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