A Dynamical Approach to the Asymptotic Behavior of the Sequence $\Omega(n)$

TL;DR

This paper investigates the asymptotic properties of the prime factor counting function (n) using dynamical systems, revealing that classical ergodic theorems do not apply along this sequence and that prime number behaviors are dynamically uncorrelated.

Contribution

It introduces a dynamical perspective on (n), showing the failure of pointwise ergodic theorems and the disjointness of prime number theorems in this context.

Findings

Operators T^{(n)} have the strong sweeping-out property.

Pointwise Ergodic Theorem does not hold along (n).

Dynamical correlations between Prime Number Theorem and Erd
51s-Kac Theorem tend to zero.

Abstract

We study the asymptotic behavior of the sequence $\{\Omega(n) \}_{ n \in \mathbb{N} }$ from a dynamical point of view, where $\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \mathcal{B}, \mu, T)$, the operators $T^{\Omega(n)}: \mathcal{B} \to L^1(\mu)$ have the strong sweeping-out property. In particular, this implies that the Pointwise Ergodic Theorem does not hold along $\Omega(n)$. Second, we show that the behaviors of $\Omega(n)$ captured by the Prime Number Theorem and Erd\H{o}s-Kac Theorem are disjoint, in the sense that their dynamical correlations tend to zero.