Effective bounds on Euler's totient

Question

Quick question: It's known that $$\limsup\frac{n}{\varphi(n)\log\log n}=e^\gamma$$ but are there known C and N such that $$\varphi(n)>\frac{Cn}{e^\gamma\log\log n}$$ for all $n>N$?

Failing that, what are good effective bounds on $\varphi$? The square root bound isn't good enough for me.

Answer 1

Yes. Look at http://en.wikipedia.org/wiki/Euler's_totient_function#Inequalities:

$$\varphi(n)>\frac{n}{e^\gamma\log\log n + \frac{3}{\log\log n}}$$ for $n>2$.