Sum of Reciprocals of Powers as Euler Product/Proof 2

Proof
From Sum of Geometric Sequence:
 * $\dfrac 1 {1 - p^{-z} } = 1 + \dfrac 1 {p^z} + \dfrac 1 {p^{2 z} } + \cdots$

From Convergence of P-Series:
 * $\ds \sum_{n \mathop = 1}^\infty n^{-z}$ is absolutely convergent


 * $\cmod z \gt 1$
 * $\cmod z \gt 1$

Thus:

The result follows from the Fundamental Theorem of Arithmetic.