Zeta Function with prime-

By Euler product , we have:

$\zeta(s) = \prod_{p=\rm prime} \frac 1{1-p^{-s}}$

where $\zeta(\cdot)$ denotes the Riemann zeta function .

Then

$\prod_{p=\rm prime} \frac {p^4}{p^4+1} = \prod_{p=\rm prime} \frac 1{1+\frac 1{p^4}} = \prod_{p=\rm prime} \frac {1-\frac 1{p^4}}{1-\frac 1{p^8}} = \boxed{\frac {\zeta(8)}{\zeta(4)}}$