Sum of Reciprocals of Powers as Euler Product/Corollary 2/Examples/Zeta^2(2) over Zeta(4)
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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^{-2} } {1 - p^{-2} } } = \dfrac 5 2$
where the infinite product runs over the prime numbers.
Proof
| \(\ds \prod_{\text {$p$ prime} } \paren {\frac {1 + p^{-s} } {1 - p^{-s} } }\) | \(=\) | \(\ds \dfrac {\paren {\map \zeta s}^2} {\map \zeta {2s } }\) | Sum of Reciprocals of Powers as Euler Product/Corollary 2 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \prod_{\text {$p$ prime} } \paren {\frac {1 + p^{-2} } {1 - p^{-2} } }\) | \(=\) | \(\ds \dfrac {\paren {\map \zeta 2}^2} {\map \zeta 4}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {\pi^2 } 6 }^2} {\dfrac {\pi^4} {90 } }\) | Basel Problem and Riemann Zeta Function of 4 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {\dfrac {\pi^4} {36} } } {\dfrac {\pi^4} {90} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {90} {36}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 5 2\) |
$\blacksquare$