TITLE: Density of integers with many prime factors
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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.