A dynamical approach to the asymptotic behavior of the sequence

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Abstract: We study the asymptotic behavior of the sequence

{O m e g a (n)}_{n i n m a t h b b N}

from a dynamical point of view, where

O m e g a (n)

denotes the number of prime factors of

n

counted with multiplicity. First, we show that for any non-atomic ergodic system

(X, m a t h c a l B, m u, T)

, the operators

T^{O m e g a (n)} : m a t h c a l B o L^{1} (m u)

have the strong sweeping-out property. In particular, this implies that the Pointwise Ergodic Theorem does not hold along

O m e g a (n)

. Second, we show that the behaviors of

O m e g a (n)

captured by the Prime Number Theorem and ErdH{o}s-Kac Theorem are disjoint, in the sense that their dynamical correlations tend to zero.

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