A dynamical approach to the asymptotic behavior of the sequence

2022/11/29 by KAITLYN LOYD · 1 citation

paper · doi:10.1017/etds.2022.81

Abstract

Abstract We study the asymptotic behavior of the sequence \Ω (n) \_ n ∈ \mathbb N from a dynamical point of view, where Ω (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, μ , T) , the operators T^{Ω (n)}: \mathcal B → L¹(μ ) have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along Ω (n) . Second, we show that the behaviors of Ω (n) captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.

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• Transforming the Erdős-Kac theorem

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