On $Q$-multiplicative functions having a positive upper-meanvalue

Abstract

A classical approach to study properties of $Q$-multiplicative functions $f(n)$ is to associate to the mean $\frac{1}{x} \sum_{0 \le n \le x} f(n)$ the product $\prod_{0 \le j \le k} \frac{1}{q_j} \sum_{0 \le a \le q_j-1} f(aQ_j)$. We discuss its validity in the case of non-negative $Q$-multiplicative functions $f(n)$ with a positive upper meanvalue, defined via a Cantor numeration system.