Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

Abstract

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 Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. 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 which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases.

56 pages, implemented changes following the referees comments

References in corpus (3)

• Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
• A new elementary proof of the Prime Number Theorem
• An extension of Weyl's equidistribution theorem to generalized polynomials and applications

Cited by in corpus (8)

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• Dynamics on the number of prime divisors for additive arithmetic semigroups
• Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
• The prime number theorem over integers of power-free polynomial values
• Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets
• On averages of completely multiplicative functions over co-prime integer pairs
• A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results
• Some ergodic theorems involving Omega function and their applications