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

The {\em Liouville function} is defined by $\text{gl}(n):=(-1{)}^{\mathrm{\Omega }(n)}$ where $\mathrm{\Omega }(n)$ is the number of prime divisors of $n$ counting multiplicity. Let ${\text{z}}_m:=e^{2\pi i\mathrm{/}m}$ be a primitive $m$–th root of unity. As a generalization of Liouville’s function, we study the functions ${\text{gl}}_{m,k}(n):={\text{z}}_m^{k\mathrm{\Omega }(n)}$. Using properties of these functions, we give a weak equidistribution result for $\mathrm{\Omega }(n)$ among residue classes. More formally, we show that for any positive integer $m$, there exists an $A>0$ such that for all $j=0,1,\mathrm{.}\mathrm{.}\mathrm{.},m-1,$ we have #\{n\leq x:\Omega(n)\equiv j (\bmod m)\}=\frac{x}{m}+O(\frac{x}{\log^A x}). Best possible error terms are also discussed. In particular, we show that for $m>2$ the error term is not $o(x^{\text{ga}})$ for any $\text{ga}<1$.