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?