Sum of Reciprocals of Powers as Euler Product/Corollary 1

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}$