an:03979181 Zbl 0606.10043 Indlekofer, K.-H. Über verallgemeinerte Momente additiver Funktionen. (On generalized moments of additive functions) DE Monatsh. Math. 103, 121-132 (1987). 0026-9255 1436-5081 1987
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11K65 11N37 generalized moments; additive functions Characterizations of additive functions f are given, for which $\| \phi \circ | f| \|:=\limsup_{x\to \infty}(1/x)\sum_{n\leq x}\phi ( | f(n)|)$ is bounded, where $$\phi: {\mathbb{R}}^+\to {\mathbb{R}}^+$$ is monotone and (1) $$\phi (x+y) \ll \phi (x)+\phi(y)$$ $$(x,y\geq 0)$$ or (2) $$\phi (x)=c^ x$$ $$(x\in {\mathbb{R}})$$. (A typical example is $$\phi (x)=x^{\alpha}$$ $$(\alpha >0)$$ for $$x\geq 0.)$$ The main result is the following theorem. Let $$f: {\mathbb{N}}\to {\mathbb{R}}$$ be additive and $$\phi(y)\uparrow \infty$$ as $$y\to \infty$$. Further, assume that (1) (or (2)) holds. Then $$\| \phi \circ | f| \| <\infty$$ if and only if the series $\sum_{p,\quad | f(p)| >1}p^{-1},\quad \sum_{p,\quad | f(p)| \leq 1}| f(p)|^ 2 p^{-1},$ $\sum_{p}\sum_{m\geq 1,\quad | f(p^ m)| >1}\phi (| f(p^ m)|) p^{-m}$ converge and $\sum_{p\leq x,\quad | f(p)| \leq 1}f(p) p^{-1} = O(1)\text{ as } x\to \infty.$