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

Abstract

Let $\Omega \left(n\right)$ denote the number of prime divisors of $n$ counting multiplicity. One can show that for any positive integer $m$ and all $j=0,1,\dots ,m-1$, we have

$$\#\{n\le x:\Omega (n)\equiv j(modm\left)\right\}=\frac{x}{m}+o\left(x^{\alpha }\right),$$

with $\alpha =1$. Building on work of Kubota and Yoshida, we show that for $m>2$ and any $j=0,1,\dots ,m-1$, the error term is not $o\left(x^{\alpha }\right)$ for any $\alpha <1$.