Some ergodic theorems over squarefree numbers and squarefull numbers

TL;DR

This paper develops ergodic theorems for squarefree and squarefull numbers, extending previous prime number theorems, and connects these results to classical probabilistic number theory theorems.

Contribution

It introduces invariant averages under multiplication for arithmetic functions and derives new ergodic theorems for special number sets, expanding the scope of dynamical number theory.

Findings

Ergodic theorems established for squarefree numbers

Ergodic theorems established for squarefull numbers

Connections made to Erdős-Kac, Bergelson-Richter, and Loyd theorems

Abstract

In 2022, Bergelson and Richter gave a new dynamical generalization of the prime number theorem by establishing an ergodic theorem along the number of prime factors of integers. They also showed that this generalization holds as well if the integers are restricted to be squarefree. In this paper, we present the concept of invariant averages under multiplications for arithmetic functions. Utilizing the properties of these invariant averages, we derive several ergodic theorems over squarefree numbers and squarefull numbers. These theorems have significant connections to the Erd\H{o}s-Kac Theorem, the Bergelson-Richter Theorem, and the Loyd Theorem.