Zeta of 2 as Product of Fractions with Prime Numerators

Theorem

 * $\displaystyle \zeta \left({2}\right) = \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {12} \times \dfrac {13} {12} \times \dfrac {13} {14} \times \cdots$

where $\zeta$ denotes the Riemann zeta function.

Proof
By definition of Riemann zeta function:
 * $\displaystyle \zeta \left({z}\right) = \prod_p \frac 1 {1 - p^{-z}}$

where $p$ ranges over the prime numbers.

Thus:

which is the result required.