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?



Density of integers with many prime factors
This is answered in:


*Mehrdad, Behzad; Zhu, Lingjiong*, [**Moderate and large deviations for the Erdős-Kac theorem**](http://dx.doi.org/10.1093/qmath/hav035), [ZBL06553541](https://zbmath.org/?q=an:06553541).


(can be found on [arxiv.org](https://arxiv.org/abs/1311.6180))
The paper also has an excellent bibliography, with many related results cited.


