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 Erdős-Kac Theorem are disjoint, in the sense that their dynamical correlations tend to zero.

arXiv comment19 pages