Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions

Abstract

We begin by establishing two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the prime number theorem, a theorem of Pillai and Selberg, a theorem of Erdős and Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. Then, by building on the ideas behind our ergodic results, we recast Sarnak’s Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak’s conjecture that focuses on the disjointness of actions of $(\mathbb{N},+)$ and $(\mathbb{N},\cdot )$. We substantiate this extension by providing proofs of several special cases.