## On the mean value of nonnegative multiplicative number-theoretical functions.(English)Zbl 0131.04303

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 $$M(g)=\lim_{N \to \infty} {1 \over N} \sum_{n \leq N} g(n)$$, 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 $$\varepsilon,$$ $$\sum_{n \leq p \leq N (1+\varepsilon)} {g(p) \log p \over p} \geq \delta (\varepsilon)$$ for $$N \geq N (\varepsilon)$$ with suitable $$\delta$$ $$(\varepsilon > 0)$$ and $$N(\varepsilon)$$, and prove (Theorem 2) that (i), (ii) and (iii) imply $M(g) = \prod_p \left[1 + {g(p)-1 \over p} \right].$ 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

### MSC:

 11N37 Asymptotic results on arithmetic functions

number theory
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