The question is for what integers $k$ s does the product $$ \prod_p (1+p^{-2}+p^{-3} +\dots+p^{-k})$$ have a nice closed form where the product is being taken over all prime. (the $p^{-1}$ term is suppose to be missing by the way, its not a typo)
Motivation
So in a math discord server we were discussing a problem, namely to find the sum: $$J(k) = \lim_{N\to \infty} \dfrac{1}{N} \sum_{N\geq n\geq 1} \dfrac{\sigma(n^k)}{n^k}$$ inspired By the famous case that $J(1)=\zeta(2)$ ofcourse. Now after some computations we found $J(2) = \dfrac{\zeta(2)\zeta(3)}{\zeta(4)}$ and then that $$J(k)=\lim_{N\to \infty} \dfrac{1}{N} \sum_{N\geq n\geq 1} \dfrac{\sigma(n^k)}{n^k}=\zeta(k+1) \prod_p (1+p^{-2}+p^{-3} +\dots+p^{-k})$$ The infinite limit is also nice, it is $\dfrac{\zeta(2)\zeta(3)}{\zeta(6)}$. I was wondering if there was some other values of $k$ for which this $J(k)$ had a nice closed form.