  We establish two ergodic theorems which have among their corollaries numerous
classical results from multiplicative number theory, including the Prime Number
Theorem, a theorem of Pillai-Selberg, a theorem of Erd\H{o}s-Delange, the mean
value theorem of Wirsing, and special cases of the mean value theorem of
Hal\'asz. By building on the ideas behind our ergodic results, we recast
Sarnak's M\"obius disjointness conjecture in a new dynamical framework. This
naturally leads to an extension of Sarnak's conjecture which focuses on the
disjointness of additive and multiplicative semigroup actions. We substantiate
this extension by providing proofs of several special cases.
