TITLE: How does one know the growth rate of distinct prime divisors?
QUESTION [5 upvotes]: I saw online a statement that
"It can be shown that the number of distinct prime divisors of an integer n, denoted $\omega(n)$, is such that 
$$\omega(n) = O\left(\frac{\log n}{\log \log n}\right)."$$ 
But I really have no idea how this is true,
In general I actually have very little experience proving any results about growth rates and such.
Does one need to  use some basic known results in number theory to prove it?

REPLY [3 votes]: The smallest number with $m$ distinct prime factors is the primorial $M=p_1\cdot\ldots\cdot p_m$.
By the prime number theorem $\sum_{p\leq x}\log p= x+o(x)$ and $p_m\approx m\log m$, hence in order to have $\omega(M)=m$ we need $M\geq e^{m\log m+O(m)}$ and that turns into
$$ \omega(M)=O\left(\frac{\log M}{\log\log M}\right) $$
as wanted.