My question is whether (*) below can be shown using the Erdős-Kac theorem? I don't think the distinction between $\Omega$ and $\omega$ is important here.
For lack of better notation let $\lambda_{r,s}(i) := 1$ if $ \Omega(i)\equiv s~ \text{(mod r)}$ and zero otherwise. For $ (0\leq s,\hat{s}< r)$ and $s\neq \hat{s},$
$$(*)\hspace{15mm} \sum_{i\leq n} \lambda_{r,s}(i) \sim \sum_{i\leq n} \lambda_{r,\hat{s}}(i) $$
Briefly the E-K theorem says that
$$\frac{\omega(n)-\log\log n}{\sqrt{\log\log n}}$$
is normally distributed. So the idea would be that for large n the area under the gaussian curve can be divided into bands (mod $r$) and the sums of areas for each $s ~\text{(mod r)}$ are roughly equal, with the caveat that for $r > 2$ the numbers have to be very large for this to work.
For $r = 3,~ n = 2^{22}$ we already have for $s \equiv 0,2,1,$
$\sum_i \lambda_{3,0}(i)/n = 0.3320,$
$\sum_i \lambda_{3,2}(i)/n = 0.3505, $
$\sum_i \lambda_{3,1}(i)/n = 0.3174, $
Note also: $r$ is fixed as n grows.