Riemann Zeta Function of 4/Proof 3

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 $

$\approx$

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

Proof

This theorem requires a proof.
In particular: $\ds \int_0^1\int_0^1\int_0^1\int_0^1 \frac 1 {1 - x_1x_2x_3x_4} \rd x_1 \rd x_2 \rd x_3 \rd x_4$. Use Beukers-Calabi-Kolk sub.
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