What is the order of the product of $ \frac{p-1}{p} $ under the square root of a prime?

Question

Is there any known asymptotical order for

$$ \prod_{p_k\ \text{prime}}^{\sqrt{p_n}} \frac{p_k-1}{p_k} $$

Answer 1

Giving the third theorem of Mertens (thank you Daniel) :

$$ \prod_{p\le n}\left(1-\frac1p\right) \sim \frac{e^{-\gamma}}{\ln n} $$

I got :

$$ \prod_{p\leq \sqrt{p_n}}\left(1 -\frac{1}{p}\right)\sim \frac{e^{-\gamma}}{\ln \sqrt p_n} \sim \frac{2e^{-\gamma}}{\ln p_n} \sim \frac{2e^{-\gamma}}{\ln (n \ln n)} $$ we know that : $$ p_n\sim nln(n)$$