Euler product for Riemann zeta and analytic continuation

Question

the Euler product for the Riemann zeta

$$ \zeta (s)= \prod _{p}\left( \sum_{n=0}^{\infty}p^{-ns}\right) $$

this is only valid for $ \Re(s) >1 $ however we could use the Borel transform so

$$ \zeta(s)=\prod_{p} \prod_{k=1}^{\infty}\int_{0}^{\infty}e^{-x_{k}}e^{-p^{-s}x_{k}}\,\mathrm{d}x_{k}$$

so if we used the Borel transform could we then expand or analytic continue the Euler product to $ 0<\Re(s)<1 $

Answer 1

I dont think, for a very simple argument: If it was the case you would have demonstrated that in the critical strip Zeta is equal to its Euler product, which is not the case !