can the Euler product be defined on the critical line $ 1/2+it $ ??
i mean the Euler product for the Riemann Zeta function $$ \zeta(s)= \prod _{n}(1-p^{-s})^{-1} $$
for example from the representation of the Dirichlet generating function for the Rieman zeta would we have that ??
$$ \prod _{n\le x}(1-p^{-0.5-it})^{-1} =\exp\left(\sum_{n \le x}\frac{\Lambda(n)}{n^{0.5+it}}\right)$$
