<s>[INST] Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$.  For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"
Are there any nice ways to prove that $$\zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}?$$
I already know some proofs that give all values of $\zeta(n)$ for positive even integers $n$ (like #7 on Robin Chapman's list or Qiaochu Yuan's answer in the linked question).  I'm not so much interested in those kinds of proofs as I am those that are specifically for $\zeta(4)$.
I would be particularly interested in a proof that isn't an adaption of one that $\zeta(2) = \pi^2/6$.
 [/INST] Consider the contour integral
$$
\oint_C\frac{\pi\cot\pi z}{z^4}\ dz
$$
where $C$ is the counter-clockwise square contour centered at origin with vertices $\left(N+\frac12\right)(\pm1\pm i)$.

Lemma

Suppose that the function $\phi(z)$ is holomorphic at $z=n\in\mathbb{Z}$ with $\phi(n)\neq0$, then $\pi\phi(z)\cot\pi z$ has a simple pole at $n$ with residue $\phi(n)$.

Proof
Note that $\tan\pi z$ have simple zeros at $z=n$, hence $\pi\phi(z)\cot\pi z$ have simple poles there and
$$
\text{Res}\left[\pi\phi(z)\cot\pi z\ ;\ n\right]=\text{Res}\left[\frac{\pi\phi(z)}{\tan\pi z}\ ;\ n\right]=\frac{\pi\phi(n)}{\pi\sec^2\pi n}=\phi(n).
$$

Thus, by residue theorem for $z\neq0$ we obtain
$$
\sum_{n=-N}^N\text{Res}\left[\frac{\pi\cot\pi z}{z^4}\ ;\ z=n\right]=\sum_{n=-N,\ n\neq0}^N\frac1{n^4}=2\sum_{n=1}^N\frac1{n^4}.\tag1
$$
From the Taylor series of $\cot\pi z$ at $z=0$ we obtain
\begin{align}
\frac{\pi\cot\pi z}{z^4}&=\frac\pi{z^4}\cos\pi z\csc\pi z\\
&=\frac\pi{z^4}\left(1-\frac{(\pi z)^2}{2!}+\frac{(\pi z)^4}{4!}-\frac{(\pi z)^6}{6!}+\cdots\right)\left(\frac1{\pi z}+\frac{\pi z}{6}+\frac{7(\pi z)^3}{360}+\cdots\right)\\
&=\frac1{z^5}\left(1-\frac{(\pi z)^2}{2!}+\frac{(\pi z)^4}{4!}-\frac{(\pi z)^6}{6!}+\cdots\right)\left(1+\frac{(\pi z)^2}{6}+\frac{7(\pi z)^4}{360}+\cdots\right)\\
\end{align}
Expanding the series above, we see that
$$
\text{Res}\left[\frac{\pi\cot\pi z}{z^4}\ ;\ z=0\right]=-\frac{\pi^4}{2!\cdot6}+\frac{\pi^4}{4!}+\frac{7\pi^4}{360}=-\frac{\pi^4}{45}.\tag2
$$
Observe that at any point on the boundary, we have
$$
\left|\frac{\pi\cot\pi z}{z^4}\right|\le\frac{\pi\coth\frac\pi2}{\left(N+\frac12\right)^4}.\tag3
$$

Proof
Putting $z=x+iy$ and using the trigonometric sum formulas and basic identities, we have
$$
|\cot\pi z|^2=\left|\frac{\cos\pi z}{\sin\pi z}\right|=\frac{\sinh^2\pi y+\cos^2\pi x}{\cosh^2\pi y-\cos^2\pi x}.
$$
On the vertices sides of contour $C$, we have $x=\pm\left(N+\frac12\right)$ giving $\cos\left(N+\frac12\right)\pi=0$, hence
$$
|\cot\pi z|=|\tanh\pi y|\le1.
$$
On the horizontal sides we have $0\le\cos^2\pi x\le1$, hence
$$
|\cot\pi z|^2\le\frac{\sinh^2\pi y+1}{\cosh^2\pi y-1}=\frac{\cosh^2\pi y}{\sinh^2\pi}=\coth^2\pi y.
$$
Therefore
$$
|\cot\pi z|\le\coth\pi y=\coth\left(N+\frac12\right)\pi\le\coth\frac\pi2
$$
Thus, on the boundary of contour $C$ we have
$$
|\cot\pi z|\le\max\left[1,\coth\frac\pi2\right]=\coth\frac\pi2
$$

From $(3)$ and the property
$$
\left|\int_C f(z)\ dz\right|\le ML,
$$
where $M$ is $\max|f(z)|$ on C and $L$ is the length of $C$, we obtain
$$
\oint_C\frac{\pi\cot\pi z}{z^4}\ dz\le\frac{\pi\coth\frac\pi2}{\left(N+\frac12\right)^4}\cdot8\left(N+\frac12\right)=\frac{8\pi\coth\frac\pi2}{\left(N+\frac12\right)^3}\to0
$$
as $N\to\infty$. Thus, using $(1)$ and $(2)$ also using residue theorem we obtain
\begin{align}
\lim_{N\to\infty}\frac{1}{2\pi i}\oint_C\frac{\pi\cot\pi z}{z^4}\ dz&=0\\
\sum_{n=-\infty}^\infty\text{Res}\left[\frac{\pi\cot\pi z}{z^4}\ ;\ z=n\right]+\text{Res}\left[\frac{\pi\cot\pi z}{z^4}\ ;\ z=0\right]&=0\\
2\sum_{n=1}^\infty\frac1{n^4}-\frac{\pi^4}{45}&=0\\
\large\color{blue}{\sum_{n=1}^\infty\frac1{n^4}}&\large\color{blue}{=\frac{\pi^4}{90}}.
\end{align}
</s>