Is there a closed form of $\prod_{p \in \mathbb{P}} p^{p^{-k}}$ ,$k >1$?

Question

I have did a search in web to get the closed form of this product $\prod_{p \in \mathbb{P}} p^{p^{-k}}$, $k >1$ and $k$ is a real number such that I have selected topics related to Euler product and Riemann zeta function but I didn't get , any help ?

Answer 1

This function can be written as $$\prod_{p\in \mathbb{P}}p^{p^{-x}} =e^{-P'(x)}$$ Where $P(x)$ is the Prime Zeta function. $$P(x)=\sum_{p\in \mathbb{P}}p^{-x} \implies P'(x)= -\sum_{p\in \mathbb{P}} p^{-x} \ln(p)$$

I don't believe we have closed forms for any specific values of prime zeta function. See here. And I would guess we don't have closed forms for $P'(x)$ either which would mean we don't have closed forms for the product you've presented.