Mertens theorem
= Mertens theorem
{c}
{wiki=Mertens'_theorems}
One form of Mertens' theorem is
$$
\prod_{p\leq x}\left(1-\frac1p\right)
\sim\frac{e^{-\gamma}}{\log x}.
$$
Taking a ratio gives $\prod_{w\leq p\leq z}(1-1/p)\ll\log w/\log z$ for $2\leq w\leq z$.