Document Zbl 0131.04303 - zbMATH Open

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.

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