Riemann Zeta as Euler Product
As you likely know, the Riemann Zeta function, commonly expressed as a sum:
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
can be written as an Euler product formula:
$$\zeta(s) = \prod_{p\ prime} \frac{1}{1-p^{-s}}$$
Zeros
For the Riemann Zeta function to be zero, this means at least one of those product factors must be zero:
$$\frac{1}{1-p^{-s}} = 0$$
Flaw?
Why is the above logic considered incorrect? Where is the flaw?
If it is indeed correct, then why don't mathematicians talk about proving the Riemann Hypothesis that non-trivial zeros lie on the $Re(s) = \frac{1}{2}$ line, by focussing on these factors, one of which must be zero?