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.



Effective bounds on Euler's totient
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 $n>2$.


