ProofWiki:Sandbox/Template

Theorem

The Riemann zeta function of $4$ is given by:

$\ds \map \zeta 4$

$=$

$\ds \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots$

$\ds $

$=$

$\ds \dfrac {\pi^4} {90}$

$\ds $

$\approx$

$\ds 1 \cdotp 08232 \, 3 \ldots$

$\ds \map \zeta 4$

$=$

$\ds \paren{\map \zeta 2 }^2 - 2 \dfrac { \pi^4} {5!}$

Squaring Zeta of 2 produces Zeta of 4 plus two times the sum associated with the 4th power term in the sin(x)/x expansion

$\ds $

$=$

$\ds \dfrac { \pi^4} {36} - \dfrac { \pi^4} {60}$

simplifying

$\ds $

$=$

$\ds \dfrac {\pi^4} {90}$

simplifying

$\blacksquare$