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Theorem

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

\(\displaystyle \map \zeta 4\) \(=\) \(\displaystyle \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\pi^4} {90}\)
\(\displaystyle \) \(\approx\) \(\displaystyle 1 \cdotp 08232 \, 3 \ldots\)


Proof

\(\displaystyle \map \zeta 4\) \(=\) \(\displaystyle \paren{\map \zeta 2 }^2 - 2 \dfrac { \pi^4} {5!}\) 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
\(\displaystyle \) \(=\) \(\displaystyle \dfrac { \pi^4} {36} - \dfrac { \pi^4} {60}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\pi^4} {90}\) simplifying

$\blacksquare$