TITLE: Number of distinct factors
QUESTION [13 upvotes]: Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb N:\mbox{ }m<n,\mbox{ }\omega(m)<(\log\log m)^c\big\}.$$
What is known about cardinality $|S(n,c)|$?
What probabilistic model has a good fit for $\omega(m)$?
I am looking for finer information such as:
How does $\frac{\large\partial\big(\big|S(n,c)\big|\big)}{\large\partial c}$ behave? 
For a given $a>1$, is there a $c$ as a function of $m$ where $1-\frac 1a$ fraction of numbers less than $m$ lie in?

REPLY [10 votes]: Let me supplement the answer of i707107 for $c<1$, i.e. when we count integers with very few prime factors. Writing 
$$ \pi_k(n):=|\{m\in\Bbb N:\mbox{ }m\leq n,\mbox{ }\omega(m)=k\}|,$$
the cardinality $|S(n,c)|$ can be well approximated with the sum of $\pi_k(n)$ for $k<(\log\log n)^c$. Now, we have very precise information for $\pi_k(n)$ in this range, in fact even in the wider range $1\leq k\leq A\log\log n$ with $A>0$ fixed. Namely, Selberg (1954) proved that
$$\pi_k(n)=\frac{n}{\log n}\frac{(\log\log n)^{k-1}}{(k-1)!}
\left\{\lambda\left(\frac{k-1}{\log\log n}\right)+O_A\left(\frac{k}{(\log\log n)^2}\right)\right\}, $$
where
$$ \lambda(z):=\frac{1}{\Gamma(z+1)}\prod_p\left(1+\frac{z}{p-1}\right)\left(1-\frac{1}{p}\right)^z.$$
You can read the proof in Chapter II.6 of Tenenbaum: Introduction to analytic and probabilistic number theory. The notes to this chapter contain further information, e.g. an extension of the above result valid for $k\ll(\log\log n)^2$. This extension, with a slightly different looking asymptotic, is due to Hildebrand-Tenenbaum (1988), and it corresponds to $c\leq 2$ in your notation.