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Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that

$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)^2+c_{-1}x(\log\log x)+c_0 x+\sum_{j=1}^{k} c_j\frac{x}{(\log x)^j}+O\left(\frac{x}{(\log x)^{k+1}}\right).$$

Somehow I have the feeling this is a well known basic result about $\omega(n)$ (basically, an improvement of a consequence of the Erdos-Kac's theorem). Does anybody know a reference for such an expansion?

Thank you very much for your help.

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    $\begingroup$ If it's out there I am guessing it'll either be in Hardy-Wright or Montgomery-Vaughan $\endgroup$ Jun 20, 2019 at 17:09
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    $\begingroup$ In "Second-order terms for the variances and covariances of the number of prime factors—including the square free case" (Journal of Number Theory, 1977), Diaconis, Mosteller and Onishi deal with $k=0$. One can use the Delange-Selberg method, as in Chapter II.6 of Tenenbaum's book, to go general $k$, but I do not know if this was written explicitly anywhere. $\endgroup$ Jun 20, 2019 at 18:07
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    $\begingroup$ @OfirGorodetsky thank you for your comment, but the case $k=0$ it's quite immediate to deduce. Moreover, I believe that using the asymptotic expansion for $\pi_k(x)$ contained in Tenenbaum's book could lead to complications. Probably the best way to deduce the above asymptotic it's simply to insert the definition of $\omega(n)$ in the sum and then estimate the various contributions from the deriving sums over primes. $\endgroup$ Jun 20, 2019 at 18:41
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    $\begingroup$ Diaconis proved a similar result for $\sum \omega(n)$; it was improved in this paper by Eric Naslund, which you could use to start sleuthing for a potential analogous result for $\sum \omega(n)^2$. $\endgroup$ Jun 21, 2019 at 3:26
  • $\begingroup$ @GregMartin thank you for your reference, the Naslund's paper is very interesting. Basically, I proved the above asymptotic for $\sum \omega(n)^2$ in exactly the same way he proved that for $\sum \omega(n)$, i.e. inserting the definition of $\omega(n)$ and computing the sum over primes of certain fractional parts using the PNT. Indeed, the result I obtained is stronger than that I wrote above and allows for an error term of the classic PNT form if we take into account the integrals $li_f$, for suitable functions $f$, in the asymptotic. $\endgroup$ Jun 21, 2019 at 9:04

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