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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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