Über verallgemeinerte Momente additiver Funktionen

Abstract

In this paper we give characterizations of additive functionsf, for which

$$\mathop {\lim \sup }\limits_{x \to \infty } x^{ - 1} \sum\limits_{n \leqslant x} {\varphi (|f(n)|)}$$

is bounded, where φ: ℝ⁺ → ℝ⁺ is monotone and

or

$$\begin{array}{*{20}c} {\varphi (x) = c^x } & {(x \in \mathbb{R}).} \\ \end{array}$$

A typical example is φ (x)=x ^a (a>0) forx≥0.