Estimating size of partial euler product

Question

What estimates are there for product over primes $p \leq x$

$\prod_{p \leq x}(1-\frac{1}{p^{r}})$

given $r$ is positive integer.

Something better than

$\prod_{p \leq x}(1-\frac{1}{p^{r}}) \leq 1-\frac{1}{x^r}$ (when $x \geq 2$)

Answer 1

For $r=1 $ we have the Merten's third theorem $$\prod_{p\leq x}\left(1-\frac{1}{p}\right)\sim\frac{e^{-\gamma}}{\log\left(x\right)} $$ for $r>1 $ note that $$\prod_{p}\left(1-\frac{1}{p^{r}}\right)=\frac{1}{\zeta\left(r\right)} $$ then you can approximate the product with the reciprocal of the Zeta function at $r$.