Properties of uniformly summable multiplicative functions (Q1061161)

Let, for any \(\alpha\geq 1\), \({\mathfrak L}_{\alpha}:=\{f: {\mathbb{N}}\to {\mathbb{C}}\), \(\| f\|_{\alpha}<\infty \}\) denote the vector space of arithmetical functions with bounded seminorm \[ \| f\|_{\alpha}:=\{ \limsup_{x\to \infty}(1/x)\sum_{n\leq x}| f(n)|^{\alpha}\}^{1/\alpha}, \] and let \({\mathfrak L}^*\) be the class of uniformly summable functions: \(f\in {\mathfrak L}^*\) in case \[ \lim_{K\to \infty}\sup_{x\geq 1}(1/x)\sum_{n\leq x, | f(n)| \geq K}| f(n)| =0. \] Obviously, if \(\alpha >1\), \({\mathfrak L}_{\alpha}\subsetneqq {\mathfrak L}^*\subsetneqq {\mathfrak L}_ 1.\) In this paper the asymptotic behaviour, as \(x\to \infty\), of the sum \[ M(f,x,\alpha):=(1/x)\sum_{n\leq x}f(n) e^{2\pi in\alpha},\quad \alpha \in {\mathbb{R}}, \] for multiplicative functions \(f\in {\mathfrak L}^*\) is studied. For \(\alpha =0\) a complete description of the mean-behaviour of (1/x)\(\sum_{n\leq x}f(n)\) (x\(\to \infty)\) is obtained. This result generalizes a well-known theorem of \textit{G. Halász} [Acta Math. Acad. Sci. Hung. 19, 365-403 (1968; Zbl 0165.058)] on multiplicative functions \(| f| \leq 1\). For irrational \(\alpha\) it is shown that \(\lim_{x\to \infty}M(f,x,\alpha)=0\) for all multiplicative functions \(f\in {\mathfrak L}^*\). As applications of this, characterizations of almost periodic and almost even (multiplicative) functions with non-empty spectrum and generalizations of results of \textit{W. Schwarz} [J. Reine Angew. Math. 307/308, 418-423 (1979; Zbl 0397.10039)] on multiplicative functions \(f\in {\mathfrak L}_ 2\) are given. (Remark. The results of this paper were announced by the author at the Oberwolfach-Conference on ''Analytische Zahlentheorie'', Nov. 2-Nov. 8, 1980).