# Euler product for Riemann zeta and analytic continuation

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 $0<\Re(s)<1$

• the question IS : can we use borel transform to make an analytic continuation of the Euler product ? May 29 '14 at 10:04