Yes. Look at <a href="http://en.wikipedia.org/wiki/Euler's_totient_function#Inequalities">http://en.wikipedia.org/wiki/Euler's_totient_function#Inequalities</a>: $$\varphi(n)>\frac{n}{e^\gamma\log\log n + \frac{3}{\log\log n}}$$ for $n>2$.