Mertens third theorem

There is a constant $C>0$ such thatTaking logarithms reduces the result to the Mertens second theorem, because the terms of order $p^{-2}$ and smaller form an absolutely convergent series.

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 $1+o(1)$.