## An almost-sure estimate for the mean of generalized $$Q$$-multiplicative functions of modulus 1.(English)Zbl 1020.11006

Let $$Q_0=1$$, $$Q_{k+1}= q_kQ_k$$, $$q_k\geq 2$$, be a Cantor scale, $$Q=(Q_k)_{k\geq 0}$$, $$\mathbb Z_Q$$ the compact group $$\prod_{0\leq j<k} \mathbb Z/q_j\mathbb Z$$, and $$\mu$$ its Haar measure. To an element $$x=(a_0,a_1,\dots)$$ of $$\mathbb Z_Q$$, $$0 \leq a_k<q_{k+1}$$, is associated the sequence $$x_k=\sum_{0\leq j\leq k}a_jQ_j$$, $$k \geq 0$$. The author characterizes the $$Q$$-multiplicative functions $$g:\mathbb N \to \mathbb C$$ of modulus 1, such that $\lim_{x_k \to x}\left( {1\over x_k}\sum_{n<x_k} g(n)-\prod_{0\leq j\leq k}{1\over q_j}\sum_{0\leq a<q_j} g(aQ_j)\right)= 0\quad\mu-\text{a.e}.$

### MSC:

 11K65 Arithmetic functions in probabilistic number theory 11N56 Rate of growth of arithmetic functions
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### References:

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