Some ergodic theorems over $k$-full numbers

TL;DR

This paper extends key ergodic theorems related to prime and squarefree numbers to the broader class of $k$-full numbers, generalizing foundational results in number theory and dynamical systems.

Contribution

It derives analogues of Bergelson-Richter, Erdős-Kac, and Loyd's theorems for $k$-full numbers, expanding the scope of ergodic theorems in number theory.

Findings

Established ergodic theorems over $k$-full numbers for all $k \u2265 2$

Generalized prime number theorem to $k$-full numbers

Extended disjoint forms of classical theorems to broader number classes

Abstract

In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem. Later, Loyd showed a disjoint form with the Erd\H{o}s-Kac theorem. Recently, the author and his coauthors proved some ergodic theorems over squarefree numbers related to these results. In this paper, building on the previous work, we will derive the analogues of Bergelson-Richter's theorem, Erd\H{o}s-Kac theorem and Loyd's theorem over $k$-full numbers for any integer $k\geq2$.