Proof of the Euler product formula for the Riemann zeta function

The Euler product formula expresses the Riemann zeta function \(\zeta(s)\) as an infinite product over all prime numbers. Specifically, it states that for \(\text{Re}(s) > 1\): \[ \zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} \] where \(p\) varies over all prime numbers.