Sum of Reciprocals of Powers as Euler Product

Theorem
The following definitions of the Riemann zeta function are equivalent on the domains on which each of the definitions hold:

Definition 2
That is:
 * $\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z}}$

for $z \in \C$ such that $\Re \left({z}\right) > 1$.