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.
2 Answers
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The original references are:
- S. Selberg [Math. Z. 44 (1939), 306–318; zbMATH:0019.39308]
- 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.
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$\begingroup$ ah, @GHfromMO -- thanks for the edit! Are you sure the metadata in DigiZeitschriften for Selberg's paper is correct? zbMATH seems to have the year of publication as 1938, and so do all references to that paper in MathSciNet (which does not seem to have indexed it). $\endgroup$ Commented Feb 14 at 22:47
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1$\begingroup$ If you open the link in my post, you will see that the journal issue containing Selberg's paper was published in 1939. You can see the original cover of that issue. $\endgroup$ Commented Feb 14 at 22:56
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$\begingroup$ Aha, thanks! Very strange that zbMATH has the wrong year... $\endgroup$ Commented Feb 14 at 22:59
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2$\begingroup$ @LSpice Delange is correct. I fixed that in the post. $\endgroup$ Commented Feb 14 at 23:55
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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).