We show that the fractional parts of n1/ω(n), n 1/Ω(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 Ω(n) and ω(n) denote the number of distinct prime divisors of n counted with and without multiplicities. Note that n1/Ω(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.
|Number of pages||9|
|Journal||Boletin de la Sociedad Matematica Mexicana|
|Publication status||Published - Oct 2006|