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For a positive integer $n$ put $\omega(n)$ for the number of distinct prime divisors of $n$. It is a well-known theorem of Erdős and Kac that the probability distribution for the quantity

$\displaystyle \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}}$

is the standard normal distribution. In other words, we have

$$\displaystyle \lim_{X \rightarrow \infty} \left(\frac{1}{X} \# \left\{n \leq X : a \leq \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \leq b \right\}\right) = \frac{1}{\sqrt{2\pi}} \int_a^b e^{-t^2/2}dt.$$

My question is, can one give a good estimate for the density of integers which deviates from the mean significantly? The above limit is only sensitive to positive density, where I am expecting a 0-density result. More precisely and concretely, how does one estimate the density of the set

$$\displaystyle \{n \leq X : \omega(n) > (\log \log n)^2\}$$ say?

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    $\begingroup$ Already Selberg's paper will allow one to get such results. See also the work of Hildebrand and Tenenbaum in Duke which gives asymptotic formulae in wide ranges of $\omega(n)$. If you just want an upper bound, a simple Rankin's trick argument might suffice. $\endgroup$ – Lucia Oct 26 '17 at 15:55
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This is answered in:

Mehrdad, Behzad; Zhu, Lingjiong, Moderate and large deviations for the Erdős-Kac theorem, ZBL06553541.

(can be found on arxiv.org) The paper also has an excellent bibliography, with many related results cited.

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  • $\begingroup$ See also P. Erdös and J-L. Nicolas, Sur la fonction nombre de facteurs premiers de n, Séminaire Delange-Pisot-Poitou. Théorie des nombres 20 (1978-1979), no. 2, 1–19 for a result (of Delange) on the number of integers up to $x$ with about $\gamma\log\log x$ prime factors for $\gamma>0$, $\gamma\ne1$. $\endgroup$ – Greg Martin Oct 26 '17 at 23:03
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As a survey in book form I would recommend Tenenbaum's book (Introduction to analytic and probabilistic number theory), chapter II. 6.1 (Integers having $k$ prime factors). Also the notes of the end of the chapter give very useful references (such as the Hildebrand-Tenenbaum paper mentioned by Lucia, the Selberg-Delange method etc.). I would doubt that extending the large deviation techniques from $\omega(n)$ about $\log \log n$ to say $(\log \log n)^2$ is of great use. These end of chapter notes rather direct to Hildebrand-Tenenbaum.

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