Riemann Zeta Function of 4/Proof 1

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

By Fourier Series of Fourth Power of x, for $x \in \closedint {-\pi} \pi$:

$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map \cos {n \pi} \map \cos {n x}$

Setting $x = \pi$:

\(\ds \pi^4\)

\(=\)

\(\ds \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map {\cos^2} {n \pi}\)

\(\ds \leadsto \ \ \)

\(\ds \frac {4 \pi^4} 5\)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2} {n^4} - \sum_{n \mathop = 1}^\infty \frac {48} {n^4}\)

Cosine of Multiple of Pi

\(\ds \leadsto \ \ \)

\(\ds \frac {\pi^4} 5\)

\(=\)

\(\ds 2 \pi^2 \sum_{n \mathop = 1}^\infty \frac 1 {n^2} - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\)

\(\ds \)

\(=\)

\(\ds \frac {\pi^4} 3 - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\)

Basel Problem

\(\ds \leadsto \ \ \)

\(\ds 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\)

\(=\)

\(\ds \frac {\pi^4} 3 - \frac {\pi^4} 5\)

rearranging

\(\ds \)

\(=\)

\(\ds \frac {2 \pi^4} {15}\)

\(\ds \leadsto \ \ \)

\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\)

\(=\)

\(\ds \frac {\pi^4} {90}\)

$\blacksquare$