Asymptotics of product $(1-1/p)$ over all primes.

Question

I am looking to solve the following problem:

Show that there exists a positive real number $A>0$ such that $$ \prod_{p \text{ prime}\\\;\; p\leq x} \left( 1 − \frac{1}{p} \right) = \frac {A}{\log x} ( 1 + O( \frac{1}{\log x} ) ) \quad \text{ as }\space x\rightarrow \infty $$

I have been trying to get this result using Mertens second theorem but I have not been successful any help?

Answer 1

Taking the $\log$ it is $$\sum_{p\le x}\log(1-p^{-1})=B+O(x^{-1})+\sum_{p\le x}p^{-1}=B+O(x^{-1})+\log\log x+M+O(1/\log x)$$ where the last step is the strong form of Mertens theorem

Exponentiating the RHS and using that $$\exp(O(1/\log x))=1+O(1/\log x)$$ gives the result.