Asymptotic growth of $\frac{n}{\phi(n)}$.

Question

I need to prove that for $n \ge 3$

$\frac{n}{\phi(n)}=O(\mathrm{loglog}n)$

But I don't know how to start, I'll be grateful if you could give me some hints. Thanks.

Answer 1

If $n=p^a$ then you can compute easily $\varphi(n)$ and $n/\varphi(n)$, and $a$ is a kind of log. If $n$ is arbitrary, use the fact that $\varphi$ is multiplicative. This is not the final answer, but it is a start..