Reference request - Pillai-Selberg Theorem

Question

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. That is, $\Omega(n)$ is even $50\%$ of the time, a multiple of $3$ $1/3$ of the time, and so on. The result is commonly known as the Pillai-Selberg Theorem, but I cannot find a proof of this anywhere.

Answer 1

The original references are:

1. S. Selberg [Math. Z. 44 (1939), 306–318; zbMATH:0019.39308]
2. S. S. Pillai [Proc. Indian Acad. Sci. Sect. A. 11 (1940), 13–20; zbMATH:66.0168.01, MR0001761].

Interestingly (I was not aware of this until I looked it up right now), the Selberg here is Sigmund Selberg, the older brother of the more famous Atle Selberg. Selberg's paper is in German, but Pillai's is in English.

You could also look at more modern references which might be easier to read. An application of Weyl's criterion and the (Atle!) Selberg-Delange method is what you need, see for example, this answer of mine.

Answer 2

Sigmund Selberg (1939) proved this for square-free numbers. You can find the original proof here.

For the general statement with a better error term than $o(x)$, see Addison (1957).