Basel Problem as Infinite Product

Theorem

 * $\displaystyle \dfrac {\pi^2} 6 = \prod_{p \mathop \in \mathbb P} \dfrac {p^2} {p^2 - 1}$

Proof
From Sum of Reciprocals of Powers as Euler Product:


 * $\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$

for $z \in \C$ such that $\map \Re z > 1$.

Putting $z = 2$:

The result follows from Riemann Zeta Function of $2$.