Uniformity of distribution modulo 1 of the geometric mean prime divisor

We show that the fractional parts of n(1/omega(n)), n(1/Omega(n)) and the geometric mean of the distinct prime factors of n are uniformly distributed modulo 1 as n ranges over all the positive integers, where Omega(n) and omega(n) denote the number of distinct prime divisors of n counted with and without multiplicities. Note that n(1/Omega(n)) is the geometric mean of all prime divisors of n taken with the corresponding multiplicities. The result complements a series of results of similar spirit obtained by various authors, while the method can be applied to several other arithmetic functions of similar structure.