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

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 12.1 (2000): 1-12. <http://eudml.org/doc/248499>.

@article{Mauclaire2000,
abstract = {Let $Q = \{(Q_k)\}_\{k \ge 0\}, Q_0 = 1, Q_\{k+1\} = q_kQ_k, q_k \ge 2$, be a Cantor scale, $\mathbf \{Z\}_Q$ the compact projective limit group of the groups $\mathbf \{Z\}/Q_k\mathbf \{Z\}$, identified to $\prod _\{0 \le j \le k-1\} \mathbf \{Z\}/q_j\mathbf \{Z\}$, and let $\mu $ be its normalized Haar measure. To an element $x = \lbrace a_0, a_1, a_2, \dots \rbrace , 0 \le a_k \le q_\{k+1\} - 1$, of $\mathbf \{Z\}_Q$ we associate the sequence of integral valued random variables $x_k = \sum _\{ 0 \le j \le k\} a_jQ_j$. The main result of this article is that, given a complex $\mathbf \{Q\}$-multiplicative function $g$ of modulus $1$, we have\begin\{equation*\}\lim \_\{x\_k \rightarrow x\} (\frac\{1\}\{x\_k\} \sum \_\{n\le x\_k-1\} g(n)- \prod \_\{0\le j \le k\} \frac\{1\}\{q\_j\} \sum \_\{0 \le a\le q\_j\} g(aQ\_j)) = 0 \quad \mu \text\{-a.e\}.\end\{equation*\}},
author = {Mauclaire, Jean-Loup},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {almost-sure estimate; mean value; -multiplicative functions},
language = {eng},
number = {1},
pages = {1-12},
publisher = {Université Bordeaux I},
title = {An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$},
url = {http://eudml.org/doc/248499},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Mauclaire, Jean-Loup
TI - An almost-sure estimate for the mean of generalized $Q$-multiplicative functions of modulus $1$
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 1
EP - 12
AB - Let $Q = {(Q_k)}_{k \ge 0}, Q_0 = 1, Q_{k+1} = q_kQ_k, q_k \ge 2$, be a Cantor scale, $\mathbf {Z}_Q$ the compact projective limit group of the groups $\mathbf {Z}/Q_k\mathbf {Z}$, identified to $\prod _{0 \le j \le k-1} \mathbf {Z}/q_j\mathbf {Z}$, and let $\mu $ be its normalized Haar measure. To an element $x = \lbrace a_0, a_1, a_2, \dots \rbrace , 0 \le a_k \le q_{k+1} - 1$, of $\mathbf {Z}_Q$ we associate the sequence of integral valued random variables $x_k = \sum _{ 0 \le j \le k} a_jQ_j$. The main result of this article is that, given a complex $\mathbf {Q}$-multiplicative function $g$ of modulus $1$, we have\begin{equation*}\lim _{x_k \rightarrow x} (\frac{1}{x_k} \sum _{n\le x_k-1} g(n)- \prod _{0\le j \le k} \frac{1}{q_j} \sum _{0 \le a\le q_j} g(aQ_j)) = 0 \quad \mu \text{-a.e}.\end{equation*}
LA - eng
KW - almost-sure estimate; mean value; -multiplicative functions
UR - http://eudml.org/doc/248499
ER -