Sum of Reciprocals of Powers as Euler Product/Corollary 2/Examples/Zeta^2(4) over Zeta(8)
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Example of Use of Sum of Reciprocals of Powers as Euler Product/Corollary 2
- $\ds \prod_{\text {$p$ prime} } \paren {\frac {1 + p^{-4} } {1 - p^{-4} } } = \dfrac 7 6$
where the infinite product runs over the prime numbers.
Proof
| \(\ds \prod_{\text {$p$ prime} } \paren {\frac {1 + p^{-s} } {1 - p^{-s} } }\) | \(=\) | \(\ds \dfrac {\paren {\map \zeta s}^2} {\map \zeta {2 s} }\) | Sum of Reciprocals of Powers as Euler Product/Corollary 2 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \prod_{\text {$p$ prime} } \paren {\frac {1 + p^{-4} } {1 - p^{-4} } }\) | \(=\) | \(\ds \dfrac {\paren {\map \zeta 4}^2} {\map \zeta 8}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {\pi^4} {90} }^2 } {\dfrac {\pi^8} {9450 } }\) | Riemann Zeta Function of 4 and Riemann Zeta Function of 8 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {\pi^8} {8100} } } {\dfrac {\pi^8} {9450} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {9450} {8100}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 7 6\) |
$\blacksquare$