Sum of Reciprocals of Powers as Euler Product/Corollary 1/Examples/Zeta(2) over Zeta(4)
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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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {90} {6 \pi^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15} {\pi^2}\) |
$\blacksquare$