Dynamical systems on arithmetic functions determined by primes

Abstract

In this paper, we study an algebra $\mathcal{A}$ consisting of all arithmetic functions, and corresponding dynamical systems acting on $% \mathcal{A}$ determined by a fixed prime $p$. Starting from free probabilistic models on $\mathcal{A}$ determined by $p$, we construct certain group dynamical systems induced by the additive group $\Bbb{R}$ of all real numbers. We investigate the basic properties and free-probabilistic data of such dynamical systems by constructing corresponding crossed product algebras.