An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$

Mauclaire, Jean-Loup

Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 1-12

Abstract (VO)

Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have

$$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

Résumé

Soit $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2,k\ge 0$ une échelle de Cantor, $Z_Q$ le groupe compact ${\prod }_{0\le j}Z/q_{j}Z,$ et $\mu$ sa mesure de Haar normalisée. A un élément $x$ of $Z_Q$ écrit $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1,k\ge 0$, on associe la suite $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. On montre que si $g$ est une fonction $\mathrm{Q}$-multiplicative unimodulaire, alors

$$\underset{x_k\to x}{lim}\left(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right)\right)=0\mu \text{-p.s}.$$