Sum of Reciprocals of Powers as Euler Product/Corollary 1/Examples/Zeta(2) over Zeta(4)

Example of Use of Sum of Reciprocals of Powers as Euler Product/Corollary 1

$\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-2} } = \dfrac {15 } {\pi^2}$

where the infinite product runs over the prime numbers.

Proof

\(\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-s} }\)

\(=\)

\(\ds \dfrac {\map \zeta s} {\map \zeta {2 s} }\)

Sum of Reciprocals of Powers as Euler Product/Corollary 1

\(\ds \leadsto \ \ \)

\(\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-2} }\)

\(=\)

\(\ds \dfrac {\map \zeta 2} {\map \zeta 4}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\dfrac {\pi^2} 6} {\dfrac {\pi^4} {90} }\)

Basel Problem and Riemann Zeta Function of 4

\(\ds \)

\(=\)

\(\ds \dfrac {90} {6 \pi^2}\)

\(\ds \)

\(=\)

\(\ds \dfrac {15} {\pi^2}\)

$\blacksquare$