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.
Reilly , Uniform weighted averages and a conjecture of B ergelson, M oreira, and R ichter
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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.