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Investigations in the theory of $$q$$-additive and $$q$$-multiplicative functions. II. (English) Zbl 1012.11008
The authors continue their investigations started in [ibid. 91, 53-78 (2001; Zbl 0980.11001)]. To formulate their new results obtained in the present paper, some definitions are needed.
Let $$q\geq 2$$ be a fixed integer, $${\mathbb A}=\{0,1,\dots,q-1\}$$, and denote by $${\mathbb N}_0$$ the set of nonnegative integers. Each $$n\in {\mathbb N}_0$$ has a unique $$q$$-ary expansion $n=\sum\limits_{j=0}^\infty \varepsilon_j(n)q^j,\;\;\;\varepsilon_j(n)\in{\mathbb A}.$ Let $$G$$ be an arbitrary Abelian group. A function $$\varphi:{\mathbb N}_0\to G$$ is called a $$G$$ valued $$q$$-additive function if $$\varphi(0)=0$$ and for every $$n\in {\mathbb N}_0$$ $\varphi(n)=\sum\limits_{j=0}^\infty \varphi(\varepsilon_j(n)q^j)$ holds. Denote by $${\mathcal A}_q(G)$$ the set of such functions. If $$G={\mathbb R}$$, then simply write $${\mathcal A}_q={\mathcal A}_q({\mathbb R})$$. Moreover, the function $$g:{\mathbb N}_0\to {\mathbb C}$$ is called $$q$$-multiplicative, if $$g(0)=1$$ and for every $$n\in {\mathbb N}_0$$ we have $g(n)=\prod\limits_{j=0}^\infty g(\varepsilon_j(n)q^j).$ The set of $$q$$-multiplicative functions is denoted by $${\mathcal M}_q$$, and the set of such functions with $$|g(n)|=1$$ $$(n\in {\mathbb N}_0)$$ by $$\bar{{\mathcal M}_q}$$. Let $$(1\leq)a_1<a_2<\dots<a_k(<q)$$ be mutually coprime integers, each of which is coprime to $$q$$. Let $$f_1,\dots,f_k\in{\mathcal A}_q$$, $$g_1,\dots,g_k\in\bar{{\mathcal M}_q}$$ and put $l(n)=f_1(a_1n)+\dots+f_k(a_kn),\;\;\;t(n)=g_1(a_1n)\dots g_k(a_kn).$
The authors give necessary and sufficient conditions for the existence of the limit distribution of $$l(n)$$, and of the mean value of $$t(n)$$.
Reviewer: L.Hajdu (Debrecen)

MSC:
 11A63 Radix representation; digital problems 11A67 Other number representations 11K65 Arithmetic functions in probabilistic number theory
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