A dynamical approach to the asymptotic behavior of the sequence

Abstract

AbstractWe 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.