Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$ where $\phi(n)$ is the euler phi function.
I was wondering if I could use something like
\begin{align} \phi(n)\geq \frac{n}{e^{\gamma}\log \log n}+O\left(\frac{n}{(\log \log n)^2}\right) \end{align}
to show this, but I'm not sure. Could anyone maybe give me some advice or tips on how to do this? I have quite a few Analytical Number Theory texts and notes with me, so even suggestions to some theorems would be great. Thanks.
The best theorem to look up is, I think, the following effective lower bound for $\phi(n)$
Theorem: For all $n>2$ we have $$ \frac{\phi(n)}{n}>\frac{1}{e^{\gamma}\log \log n+\frac{3}{\log \log n}}. $$