By Mertens' third theorem: $$\prod_{p\leq x}\dfrac{p-1}{p}\sim\dfrac{e^{-\gamma}}{\log x}$$ But does there exist a constant $C$ such that: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}\cdot C$$ Or otherwise, does there exist a (hopefully very small) constant $C$ for which: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}+ C$$ The only (trivial) thing I've found so far is: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}+ 0.5$$
Oh and if the constant exists, what is it's value? That's the main thing I would like to know.