For h from a Hardy field with polynomial growth, h(Ω(n)), h(ω(n)), and h(Ω(q_n)) are uniformly distributed mod 1 precisely when h deviates from rational polynomials according to one of two explicit growth conditions.
Loyd , A dynamical approach to the asymptotic behavior of the sequence (n) , Ergodic Theory and Dynamical Systems 43 no
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Weighted averages of arithmetic functions and applications to equidistribution
For h from a Hardy field with polynomial growth, h(Ω(n)), h(ω(n)), and h(Ω(q_n)) are uniformly distributed mod 1 precisely when h deviates from rational polynomials according to one of two explicit growth conditions.