Über verallgemeinerte Momente additiver Funktionen. (On generalized moments of 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 \[ \sum_{p\leq x,\quad | f(p)| \leq 1}f(p) p^{-1} = O(1)\text{ as } x\to \infty. \]
scientific article; zbMATH DE number 3979181
scientific article