What is the best known approximation for the finite product $1-1/p$ over the primes $\le x\ $?

Question

I know that the product

$$f(x)=\prod_{p\le x\ ,\ p\text{ prime}} 1-\frac{1}{p}$$

has the asymptotic behaviour $\ f(x)\approx \frac{e^{-\gamma}}{\ln(x)}\ $

But this approximation is not very good for small $x$ , lets say , $\ x\le 10^6\ $

What is the best known approximation for $f(x)$ , which is already good for $x\ge 10^6$ ?

I doubt that a series at $x=\infty$ is known, but maybe an empirical formula has been found.

The function $\ g(x)=\frac{0.6115}{\ln(1.7x)}-\frac{0.6734}{x^2}+\frac{2.6978}{x^4}\ $ seems to satisfy $\ |f(x)-g(x)|\le 0.0021\ $ for $\ x\ge67$

Has anyone idea how this can be proven ?

Answer 1

We have the Pierre Dusart's estimates $$\prod_{p\leq x}\left(1-\frac{1}{p}\right)<\frac{e^{-\gamma}}{\log\left(x\right)}\left(1+\frac{0.2}{\log^{2}\left(x\right)}\right),\, x>1 $$ and $$\frac{e^{-\gamma}}{\log\left(x\right)}\left(1-\frac{0.2}{\log^{2}\left(x\right)}\right)<\prod_{p\leq x}\left(1-\frac{1}{p}\right),\, x\geq2973.$$ I'm not completely sure, but I think those are the best known approximations today.