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

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.