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

Jean-Loup Mauclaire

An almost-sure estimate for the mean of generalized

Q

-multiplicative functions of modulus

1

Jean-Loup Mauclaire

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

Résumé
Abstract

Soit $Q = {(Q_{k})}_{k \geq 0}, Q_{0} = 1, Q_{k + 1} = q_{k} Q_{k}, q_{k} \geq 2, k \geq 0$ une échelle de Cantor, $Z_{Q}$ le groupe compact $\prod_{0 \leq j} Z / q_{j} Z,$ et $μ$ sa mesure de Haar normalisée. A un élément $x$ of $Z_{Q}$ écrit $x = {a_{0}, a_{1}, a_{2}, \dots}, 0 \leq a_{k} \leq q_{k + 1} - 1, k \geq 0$ , on associe la suite $x_{k} = \sum_{0 \leq j \leq k} a_{j} Q_{j}$ . On montre que si $g$ est une fonction $Q$ -multiplicative unimodulaire, alors

lim_{x_{k} \to x} (\frac{1}{x_{k}} \sum_{n \leq x_{k} - 1} g (n) - \prod_{0 \leq j \leq k} \frac{1}{q_{j}} \sum_{0 \leq a \leq q_{j}} g (a Q_{j})) = 0 μ -p.s .

Let $Q = {(Q_{k})}_{k \geq 0}, Q_{0} = 1, Q_{k + 1} = q_{k} Q_{k}, q_{k} \geq 2$ , be a Cantor scale, $𝐙_{Q}$ the compact projective limit group of the groups $𝐙 / Q_{k} 𝐙$ , identified to $\prod_{0 \leq j \leq k - 1} 𝐙 / q_{j} 𝐙$ , and let $μ$ be its normalized Haar measure. To an element $x = {a_{0}, a_{1}, a_{2}, \dots}, 0 \leq a_{k} \leq q_{k + 1} - 1$ , of $𝐙_{Q}$ we associate the sequence of integral valued random variables $x_{k} = \sum_{0 \leq j \leq k} a_{j} Q_{j}$ . The main result of this article is that, given a complex $𝐐$ -multiplicative function $g$ of modulus $1$ , we have

lim_{x_{k} \to x} (\frac{1}{x_{k}} \sum_{n \leq x_{k} - 1} g (n) - \prod_{0 \leq j \leq k} \frac{1}{q_{j}} \sum_{0 \leq a \leq q_{j}} g (a Q_{j})) = 0 μ -a.e .

MR Zbl

@article{JTNB_2000__12_1_1_0,
     author = {Jean-Loup Mauclaire},
     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},
     pages = {1--12},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {1},
     year = {2000},
     zbl = {1020.11006},
     mrnumber = {1827834},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2000__12_1_1_0/}
}

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

Bibliographie
Cité par

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