# Distribution of integers with number of prime factors lying in a given arithmetic progression

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 $m=3,a=1$?

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 $q$.
• Basically, one inverts the condition that $\Omega(n)\equiv a \mod m$ using additive characters mod $m$. The sums $\sum_{n\le x}v^{\Omega(n)}$ with $v=e^{2\pi i b/m}$ are then estimated using the Selberg-Delange method (see Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory"). Commented Jul 30, 2013 at 12:36