An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, p. 1-12

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 $𝐙/{Q}_{k}𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/{q}_{j}𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\left\{{a}_{0},{a}_{1},{a}_{2},\cdots \right\},0\le {a}_{k}\le {q}_{k+1}-1$, 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 $\underset{{x}_{k}\to x}{lim}\left(\frac{1}{{x}_{k}}\sum _{n\le {x}_{k}-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{{q}_{j}}\sum _{0\le a\le {q}_{j}}g\left(a{Q}_{j}\right)\right)=0\phantom{\rule{1em}{0ex}}\mu \text{-a.e}.$

Soit $Q={\left({Q}_{k}\right)}_{k\ge 0},{Q}_{0}=1,{Q}_{k+1}={q}_{k}{Q}_{k},{q}_{k}\ge 2,\phantom{\rule{0.166667em}{0ex}}k\ge 0$ 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 $x=\left\{{a}_{0},{a}_{1},{a}_{2},\cdots \right\},0\le {a}_{k}\le {q}_{k+1}-1,k\ge 0$, 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 $\underset{{x}_{k}\to x}{lim}\left(\frac{1}{{x}_{k}}\sum _{n\le {x}_{k}-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{{q}_{j}}\sum _{0\le a\le {q}_{j}}g\left(a{Q}_{j}\right)\right)=0\phantom{\rule{1em}{0ex}}\mu \text{-p.s}.$

@article{JTNB_2000__12_1_1_0,
author = {Mauclaire, Jean-Loup},
title = {An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {12},
number = {1},
year = {2000},
pages = {1-12},
zbl = {1020.11006},
mrnumber = {1827834},
language = {en},
url = {http://www.numdam.org/item/JTNB_2000__12_1_1_0}
}

Mauclaire, Jean-Loup. An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, pp. 1-12. http://www.numdam.org/item/JTNB_2000__12_1_1_0/

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