On the distribution modulo one of the mean values of some arithmetical functions

Florian Luca asked whether the sequence consisting of the arithmetic(extit {resp.} geometric) mean values of the Euler $ \varphi$ functionis uniformly distributed modulo $1$. We show that it is the case for the arithmetic means, and that it is the case for the geometric means if and only if the number $e^{-1} \prod_p (1-1/p)^{1/p}$ is irrational. More general arithmetic functions are also considered.