I'm fascinated by the Riemann Zeta hypothesis - I haven't yet taken a course in complex analysis but I'm curious what this sentence means on Wikipedia:
The convergence of the Euler product shows that ζ(s) has no zeros in this region, as none of the factors have zeros.
I'm wondering if a) the convergence of the Euler product comes purely from the fact that the p-series diverges when the exponent is greater than 1? b) What the sentence from Wikipedia is essentially saying - I'm not understanding what factors are being referred to or how the convergence shows that the Zeta function has no zeroes for $\Re(s) > 1$.
Edit: I was looking over this answer and it seems to be similar, but how do we know that, as this author wrote, $(1-p^{-s})^{-1}\neq 0$ for all primes $p$? Can't it converge to 0, meaning that $(1-p^{-s})^{-1} = 0$ for at least one prime $p$?