Totient lower bound from the Mertens product

If $C$ is the constant in the Mertens third theorem, thenSplit 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 $n/\phi (n)$, while the large primes contribute a factor $1+o(1)$.