Some ergodic theorems over squarefree numbers and squarefull numbers
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.
Keywords
Cite
@article{arxiv.2405.18157,
title = {Some ergodic theorems over squarefree numbers and squarefull numbers},
author = {Huixi Li and Biao Wang and Chunlin Wang and Shaoyun Yi},
journal= {arXiv preprint arXiv:2405.18157},
year = {2025}
}
Comments
21 pages. Accepted by Acta Arithmetica. The preprint arXiv:2406.15698 has been merged into this paper