Dynamics on the number of prime divisors for additive arithmetic semigroups

TL;DR

This paper extends the ergodic theory of prime divisor functions to additive arithmetic semigroups, connecting prime factor distributions with dynamical systems and generalizing classical number theory results.

Contribution

It introduces a new generalization linking the distribution of prime divisors and largest prime factors within additive arithmetic semigroups, extending prior ergodic theorems.

Findings

New generalization of Bergelson and Richter's theorem

Analogues established for arithmetic semigroups from finite fields

Connections between prime divisor counts and dynamical properties

Abstract

In 2020, Bergelson and Richter gave a dynamical generalization of the classical Prime Number Theorem, which has been generalized by Loyd in a disjoint form with the Erd\H{o}s-Kac Theorem. These generalizations reveal the rich ergodic properties of the number of prime divisors of integers. In this article, we show a new generalization of Bergelson and Richter's Theorem in a disjoint form with the distribution of the largest prime factors of integers. Then following Bergelson and Richter's techniques, we will show the analogues of all of these results for the arithmetic semigroups arising from finite fields as well.