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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.

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up vote 7 down vote accepted

Yes. Look at's_totient_function#Inequalities:

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

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