15 October 2022 Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions
Vitaly Bergelson, Florian K. Richter
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Duke Math. J. 171(15): 3133-3200 (15 October 2022). DOI: 10.1215/00127094-2022-0055

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 (N,+) and (N,). We substantiate this extension by providing proofs of several special cases.

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Vitaly Bergelson. Florian K. Richter. "Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions." Duke Math. J. 171 (15) 3133 - 3200, 15 October 2022. https://doi.org/10.1215/00127094-2022-0055

Information

Received: 12 January 2020; Revised: 14 October 2021; Published: 15 October 2022
First available in Project Euclid: 7 October 2022

MathSciNet: MR4497225
zbMATH: 1514.37018
Digital Object Identifier: 10.1215/00127094-2022-0055

Subjects:
Primary: 37A45
Secondary: 11J71

Keywords: additive function , entropy of semigroup actions , ergodic theorem , Liouville function , Möbius function , multiplicative function , nilmanifold , Prime Number Theorem , Sarnak’s conjecture , uniform distribution , uniquely ergodic system

Rights: Copyright © 2022 Duke University Press

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Vol.171 • No. 15 • 15 October 2022
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