On the Residue Class Distribution of the Number of Prime Divisors of an Integer

The Liouville function is defined by λ(n) := (−1)Ω(n) where Ω(n) is the number of prime divisors of n counting multiplicity. Let ζm := e2πi/m be a primitive m–th root of unity. As a generalization of Liouville’s function, we study the functions λm,k(n) := ζ kΩ(n) m . Using properties of these functions, we give a weak equidistribution result for Ω(n) among residue classes. More formally, we show that for any positive integer m, there exists an A > 0 such that for all j = 0, 1, . . . ,m− 1, we have #{n ≤ x : Ω(n) ≡ j (mod m)} = x m +O „ x log x « . Best possible error terms are also discussed. In particular, we show that for m > 2 the error term is not o(xα) for any α < 1.