# Zeta of 2 as Product of Fractions with Prime Numerators

## Theorem

 $\ds \map \zeta 2$ $=$ $\ds \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }$ $\ds$ $=$ $\ds \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {12} \times \dfrac {13} {12} \times \dfrac {13} {14} \times \cdots$

where:

$\zeta$ denotes the Riemann zeta function
$\ds \prod_p$ denotes the product over all prime numbers.

## Proof

$\ds \map \zeta z = \prod_p \frac 1 {1 - p^{-z} }$

where $p$ ranges over the prime numbers.

Thus:

 $\ds \map \zeta 2$ $=$ $\ds \prod_p \frac 1 {1 - p^{-2} }$ $\ds$ $=$ $\ds \prod_p \frac {p^2} {p^2 - 1}$ multiplying top and bottom by $p^2$ $\ds$ $=$ $\ds \prod_p \frac {p^2} {\paren {p - 1} \paren {p + 1} }$ Difference of Two Squares $\ds$ $=$ $\ds \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }$

which is the result required.

$\blacksquare$