Totient lower bound from the Mertens product
= Totient lower bound from the Mertens product
If $C$ is the constant in the <Mertens third theorem>, then
$$
\varphi(n)\geq\left(C^{-1}+o(1)\right)\frac n{\log\log n}.
$$
Split the prime divisors of $n$ at $y=\log n/\sqrt{\log\log n}$. The small primes contribute at most $(C+o(1))\log\log n$ to $n/\varphi(n)$, while the large primes contribute a factor $1+o(1)$.