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Some ergodic theorems over squarefree numbers and squarefull numbers

Huixi Li, Biao Wang, Chunlin Wang, Shaoyun Yi

TL;DR

The paper develops invariant averages under multiplication for bounded arithmetic functions and uses them to prove ergodic theorems over squarefree and squarefull (k-full) numbers. For squarefree numbers, the limiting mean of a(n) equals (α(𝒮)/ζ(2)) A, with α(𝒮)=∏_{p∈𝒮} p/(p+1), and reduces to (6/π^2) A when 𝒮 is empty; for k-full numbers, the normalized mean converges to c_k A, where c_k = ∏_{p}(1+∑_{m=k+1}^{2k-1} p^{-m/k}). The results connect to the Erdős–Kac framework, Bergelson–Richter generalizations, and Loyd-type asymptotic uncorrelatedness, yielding squarefree and k-full analogues of BR, EK, and Loyd theorems as well as a Richter-type generalization in these restricted settings. The work provides a unifying approach to ergodic phenomena on restricted multiplicative sets and broadens dynamical perspectives on the prime number theorem and related probabilistic number-theoretic limit laws.

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 Erdős-Kac Theorem, the Bergelson-Richter Theorem, and the Loyd Theorem.

Some ergodic theorems over squarefree numbers and squarefull numbers

TL;DR

The paper develops invariant averages under multiplication for bounded arithmetic functions and uses them to prove ergodic theorems over squarefree and squarefull (k-full) numbers. For squarefree numbers, the limiting mean of a(n) equals (α(𝒮)/ζ(2)) A, with α(𝒮)=∏_{p∈𝒮} p/(p+1), and reduces to (6/π^2) A when 𝒮 is empty; for k-full numbers, the normalized mean converges to c_k A, where c_k = ∏_{p}(1+∑_{m=k+1}^{2k-1} p^{-m/k}). The results connect to the Erdős–Kac framework, Bergelson–Richter generalizations, and Loyd-type asymptotic uncorrelatedness, yielding squarefree and k-full analogues of BR, EK, and Loyd theorems as well as a Richter-type generalization in these restricted settings. The work provides a unifying approach to ergodic phenomena on restricted multiplicative sets and broadens dynamical perspectives on the prime number theorem and related probabilistic number-theoretic limit laws.

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 Erdős-Kac Theorem, the Bergelson-Richter Theorem, and the Loyd Theorem.
Paper Structure (19 sections, 16 theorems, 97 equations, 1 table)

This paper contains 19 sections, 16 theorems, 97 equations, 1 table.

Table of Contents

  1. Introduction and statement of results
  2. Some ergodic theorems over squarefree numbers
  3. Some ergodic theorems over squarefull numbers
  4. An application to Richter's generalization of the prime number theorem
  5. Preliminaries
  6. Uniquely ergodic topological dynamical system
  7. Erdős-Kac Theorem
  8. Loyd Theorem
  9. k-full numbers
  10. Proof of Theorem \ref{['mainthm_sqfree']}
  11. Proof of Theorem \ref{['mainthm_sqfree_applications']}
  12. Proof of Theorem \ref{['mainthm_kfull']}
  13. Proof of Theorem \ref{['mainthm_kfull_applications']}
  14. Proof of
  15. Proof of
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $a: \mathbb{N}\to\mathbb{C}$ be a bounded arithmetic function of average $A$. Let $\mathcal{S}$ be a finite set of primes. If the average of $a(n)$ is invariant under multiplications, then where $\alpha(\mathcal{S})=\prod_{p\in \mathcal{S}}\frac{p}{p+1}$. In particular, if $\mathcal{S}=\emptyset$, then we have

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 2.1: BergelsonRichter2022
  • Theorem 2.2: BergelsonRichter2022
  • Theorem 2.3: Erdős-Kac Theorem
  • ...and 12 more