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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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