Notation question: $\ll$

Question

I was perusing http://mathworld.wolfram.com/HighlyCompositeNumber.html and saw the following at the end: Nicholas proved that there exists a constant $c_2>0$ such that $Q(x) \ll (\ln x)^{c_2}$. What does the $\ll$ mean in this context? (fyi, $Q(x)$ is the number of highly composite numbers less than or equal to $x$)

Answer 1

The symbols $\ll$ and $\gg$ denote "asymptotically very much less than" and "asymptotically very much greater than," respectively.

Answer 2

As mentioned above, $\ll$ and $\lll$ can refer to much less than, or very much less than respectively.

There do exist other uses for the symbols as well. In particular, in measure theory (seemingly unrelated to your specific example), for two measures $\lambda$ and $\mu$, you have that $\lambda \ll \mu$ (i.e., $\lambda$ is absolutely continuous with respect to $\mu$) if for all sets $\mu (E) = 0 \Rightarrow \lambda (E) = 0$