I have from my analytic number theory notes that states $$\prod_{p\le P} \sum_{k=0}^\infty \frac{1}{p^{ks}} = \sum_{n=1}^\infty \frac{c_P(n)}{n^s}$$ for primes $p$ and $Re(s)>1$. $c_P(n)$ is $1$ if all prime factors of $n$ is $\le P$ and $0$ otherwise.
This seems to be obtained throught swapping the finite product with the infinite sum. I think this is a general result that holds for absolutely convergent series but I can't find any proof for it. I would greatly appreciate if anyone could provide me insight into why this holds and how we can represent the rearranged infinite sum.