Sum of Reciprocals of Powers as Euler Product/Corollary 1/Examples/Zeta(4) over Zeta(8)
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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^{-4} } = \dfrac {105 } {\pi^4}$
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 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-4} }\) | \(=\) | \(\ds \dfrac {\map \zeta 4} {\map \zeta 8}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\dfrac {\pi^4 } {90} } {\dfrac {\pi^8} {9450} }\) | Riemann Zeta Function of 4 and Riemann Zeta Function of 8 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {9450} {90 \pi^4}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {105} {\pi^4}\) |
$\blacksquare$