Mertens' theorems

One form of Mertens' theorem isTaking a ratio gives ${\prod }_{w\le p\le z}(1-1/p)\ll \log w/\log z$ for $2\le w\le z$.

There is an absolute constant $B_1$, the Meissel–Mertens constant, such that

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 $1+o(1)$.