Sum of Reciprocals of Powers as Euler Product/Corollary 1
Jump to navigation
Jump to search
Corollary to Sum of Reciprocals of Powers as Euler Product
Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then:
- $\ds\prod_{\text {$p$ prime} } \paren {1 + p^{-s} } = \dfrac {\map \zeta s} {\map \zeta {2 s} }$
where the infinite product runs over the prime numbers.
Proof
\(\ds \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }\) | \(=\) | \(\ds \map \zeta s\) | Sum of Reciprocals of Powers as Euler Product | |||||||||||
\(\ds \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-2 s} }\) | \(=\) | \(\ds \map \zeta {2s }\) | ||||||||||||
\(\ds \prod_{\text {$p$ prime} } \frac {1 - p^{-2 s} } {1 - p^{-s} }\) | \(=\) | \(\ds \dfrac {\map \zeta s} {\map \zeta {2 s} }\) | ||||||||||||
\(\ds \prod_{\text {$p$ prime} } \frac {\paren {1 + p^{-s} } \paren {1 - p^{-s} } } {1 - p^{-s} }\) | \(=\) | \(\ds \dfrac {\map \zeta s} {\map \zeta {2 s} }\) | ||||||||||||
\(\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-s} }\) | \(=\) | \(\ds \dfrac {\map \zeta s} {\map \zeta {2 s} }\) |
$\blacksquare$
Examples
Example: $\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-2} }$
- $\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-2} } = \dfrac {15 } {\pi^2}$
Example: $\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-4} }$
- $\ds \prod_{\text {$p$ prime} } \paren {1 + p^{-4} } = \dfrac {105 } {\pi^4}$