On the residue class distribution of the number of prime divisors of an integer

Let Ω(n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j =0, 1, . . . , m−1, we have #{n ≤ x : Ω(n) ≡ j(modm)} = x/m + o(xα), with α = 1. Building on work of Kubota and Yoshida, we show that for m>2 and any j =0, 1, . . . , m − 1, the error term is not o(xα) for any α