For an integer $n$ with decomposition $n=p\_1^{e\_1}...p\_k^{e\_k}$ denote
$\lambda(n)=\sum e\_i$. It follows from the prime number theorem that
$\#\{n\le x|\lambda(n)=a\pmod{2}\}\equiv x/2+O(x\exp(-c\log^{1/2}x))$ (for $a=0,1$).
What is known about $\#\{n\le x|\lambda(n)\equiv a\pmod{m}\}$ for other fixed $a,m$,
say $m=3,a=1$?



Distribution of integers with number of prime factors lying in a given arithmetic progression
What you call $\lambda(n)$ is often denoted $\Omega(n)$ in the literature. 


Hubert Delange, Sur la distribution des valeurs de certaines fonctions arithmétiques, Colloque sur la Théorie des Nombres, Bruxelles, 1955, pp. 147–161, Georges Thone, Liège; Masson and Cie, Paris, 1956, MR0085291 (19,17b) gets a very general result, of which the following is a special case: the values of $\Omega(n)$ are equally distributed over the residue classes mod $q$, for integer $q$. 


