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A dynamical approach to the asymptotic behavior of the sequence $\Omega (n)$

Part of: Multiplicative number theory Ergodic theory

Published online by Cambridge University Press:  29 November 2022

KAITLYN LOYD Open the ORCID record for KAITLYN LOYD [Opens in a new window]
KAITLYN LOYD*
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
*
e-mail: loydka@math.northwestern.edu
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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.

Keywords

classical ergodic theoryergodic theoremsnumber theoryprime number theorem

MSC classification

Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 11N37: Asymptotic results on arithmetic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3685 - 3706
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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