In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then $$\frac{|\{n \le x : \frac{f(n) - A(x)}{\sqrt{B(x)}} \le z\}|}{x}$$ converges weakly to the Gaussian distribution $G(z)$, where $$A(x) = \sum_{p \le x}\frac{f(p)}{p} \text{ and } B(x) =\sum_{p \le x}\frac{f(p)^2}{p} $$ as long as $B(x) \rightarrow \infty$ as $x \rightarrow \infty$.

With some work and using results from "Sieving and the Erdos-Kac theorem" by Granville and Soundararajan, I am able to show that $$\frac{|\{n \le x, n \text{ squarefree } : \frac{f(n) - A(x)}{\sqrt{B(x)}} \le z\}|}{|\{n \le x : n \text{ squarefree }\}|}$$ converges weakly to $G(z)$ under the same hypotheses.

This seems like it should be a standard result but I haven't been able to find a reference for it. Does anyone know of a reference for this?

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    $\begingroup$ I agree that this should follow from known techniques, but I wouldn't be surprised if it had never appeared anywhere in this form. $\endgroup$ – Greg Martin Jan 4 '12 at 21:15
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    $\begingroup$ @Rodrigo, I have a definite preference for $\#\cdot$ over $|\cdot|$ for the number of elements in a finite set. The $\#$ notation is completely apt for counting, while $|\cdot|$ takes some interpretation and is easily confused with other uses of those symbols. $\endgroup$ – Greg Martin May 13 '14 at 17:58
  • $\begingroup$ I'm sorry, the previous latex was broken, so I just edited it in order to make sense. It hadn't came to my mind that the symbol $\#$ was intended to be used as the notation of counting, I thought it was just some misused latex command. Feel free to edit if you think it is ambiguous. $\endgroup$ – Rodrigo May 13 '14 at 19:24

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