Limit distributions of multiplicative functions with logarithmic moments
The following theorem is proved: Let \(g\colon \mathbb N\to\mathbb R\) be a multiplicative function and \(0<\alpha <1\). Then the following statements are equivalent: (i) \(g\) has nondegenerate limit distribution \(F\) and \(\int_{R\setminus \{0\}} | \log | g| |^\alpha \, dF<\infty\). (ii) The series \[ \sum_{g(p)=0}\frac{1}{p},\quad \sum_{g(p)\neq 0, | \log | g(p)| | \leq 1} \frac{\log | g(p)|}{p}, \] \[ \sum_{g(p)\neq 0, | \log | g(p)| | \leq 1} \frac{(\log | g(p)|)^2}{p}, \] \[ \sum_{g(p^ m)\neq 0, | \log | g(p^ m)| | >1} \frac{| \log | g(p^m)| |^\alpha}{p^m} \] converge. (iii) The series \(\sum_{g(p)\neq 0}1/p\) converge. If \[ f(p^m)=\begin{cases} \log | g(p^m)| &\text{ if } g(p^m)\neq 0, \\ 0 &\text{ if } g(p^m)=0,\end{cases} \] \[ f_1(p^m)=\begin{cases} f(p) &\text{ if } m=1\text{ and } | f(p)| \leq 1, \\ 0 &\text{otherwise},\end{cases} \] \(f\) and \(f_1\) additive functions, then \[ M(f_1){\overset{def} {=}}\lim_{x\to \infty}(1/x)\sum_{n\leq x}f_1(x) \] exists and \(M(| f|^\alpha)<\infty\). A similar theorem for \(\alpha =2\) has been proved earlier by A. Laurinčikas.
scientific article; zbMATH DE number 4023404