Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series. Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and using Parseval's identity, I have computed $\zeta(2)$ and $\zeta(4)$.
How can I compute $ \zeta(6) $ now?
Fourier series of $ f $:
$$ S(f)= \frac{\pi^2}{6}-\sum_{n=1}^{\infty} \frac{\cos(2nx)}{n^2}$$
$$ x=0,  \zeta(2)=\pi^2/6$$
 A: One method is to consider the generating function of $\zeta(2k)$:
$$
\begin{align}
f(x)
&=\sum_{k=1}^\infty\zeta(2k)\,x^{2k}\\
&=\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{x^{2k}}{n^{2k}}\\
&=\sum_{n=1}^\infty\frac{x^2/n^2}{1-x^2/n^2}\\
&=\sum_{n=1}^\infty\frac{x^2}{n^2-x^2}\\
&=-\frac{x}{2}\sum_{n=1}^\infty\left(\frac{1}{x-n}+\frac{1}{x+n}\right)\\
&=-\frac{x}{2}\left(\pi\cot(\pi x)-\frac1x\right)\\
&=\frac12(1-\pi x\cot(\pi x))\tag{1}
\end{align}
$$
In light of equation $(1)$, find the power series of
$$
x\cot(x)=\sum_{k=0}^\infty a_kx^{2k}
$$
$$
\cos(x)=\frac{\sin(x)}{x}\sum_{k=0}^\infty a_kx^{2k}
$$
$$
\begin{align}
\sum_{n=0}^\infty(-1)^n\!\frac{x^{2n}}{(2n)!}
&=\sum_{n=0}^\infty(-1)^n\!\frac{x^{2n}}{(2n+1)!}\;\;\sum_{k=0}^\infty a_kx^{2k}\\
&=\sum_{n=0}^\infty\left(\sum_{k=0}^n(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\right)x^{2n}\tag{2}
\end{align}
$$
Comparing the coefficients of the powers of $x$ in $(2)$ yields
$$
\begin{align}
a_n
&=\frac{(-1)^n}{(2n)!}-\sum_{k=1}^n(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\\
&=\frac{(-1)^n2n}{(2n+1)!}-\sum_{k=1}^{n-1}(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\tag{3}
\end{align}
$$
Since $a_n=-2\dfrac{\zeta(2n)}{\pi^{2n}}$ for positive $n$, $(3)$ becomes
$$
\zeta(2n)=\frac{(-1)^{n-1}\pi^{2n}}{(2n+1)!}n+\sum_{k=1}^{n-1}\!\frac{(-1)^{k-1}\pi^{2k}}{(2k+1)!}\zeta(2n-2k)\tag{4}
$$
Equation $(4)$ gives $\zeta(2n)$ recursively for positive $n$:
$$
\begin{align}
\zeta(2)&=\frac{\pi^2}{3!}=\frac{\pi^2}{6}\\
\zeta(4)&=-\frac{\pi^4}{5!}2+\frac{\pi^2}{3!}\zeta(2)=\frac{\pi^4}{90}\\
\zeta(6)&=\frac{\pi^6}{7!}3-\frac{\pi^4}{5!}\zeta(2)+\frac{\pi^2}{3!}\zeta(4)=\frac{\pi^6}{945}\\
\zeta(8)&=-\frac{\pi^8}{9!}4+\frac{\pi^6}{7!}\zeta(2)-\frac{\pi^4}{5!}\zeta(4)+\frac{\pi^2}{3!}\zeta(6)=\frac{\pi^8}{9450}
\end{align}
$$
A: Let $f(t):=t^3\ \ (-\pi\leq t\leq \pi)$, extended to all of ${\mathbb R}$ periodically with period $2\pi$. The Fourier series of this function is
$$t^3=\sum_{k=1}^\infty {2(-1)^{k-1}(k^2\pi^2-6)\over k^3}\sin(kt)\qquad(-\pi\leq t\leq \pi).$$
We now use Parseval's formula
$$\|f\|^2=\sum_{k=0}^\infty |c_k|^2.$$
But 
$$\|f\|^2={1\over\pi}\int_{-\pi}^\pi x^6\>dt={2\pi^6\over7}$$
and
$$c_k^2={4(k^2\pi^2-6)^2\over k^6}=4\left(\frac{36}{k^6}-\frac{12\pi^2}{k^4}+\frac{\pi^4}{k^2}\right),k\geq1.$$ 
Noting that
$$ \sum_{k=1}^\infty \frac{12\pi^2}{k^4}=12\pi^2\frac{\pi^4}{90}=\frac{2\pi^6}{15}, \sum_{k=1}^\infty\frac{\pi^4}{k^2}=\pi^4\frac{\pi^2}{6}=\frac{\pi^6}{6}$$
we have
$$ \sum_{k=1}^\infty \frac{1}{k^6}=\left({\pi^6\over14}+\frac{2\pi^2}{15}-\frac{\pi^4}{6}\right)/36=\frac{\pi^6}{945}. $$
A: I have posted here in Portuguese a recursive method based on the computation of the Fourier trigonometric series expansion for the function defined in $\left[ -\pi ,\pi \right] $ by  $f(x)=x^{2p}$ and extended to all of ${\mathbb R}$ periodically with period $2\pi.$ This is a shorter description than the original. In this reply I outline the case $\zeta(4)$. For $p=3$ the expansion is
$$x^{6}=\dfrac{\pi ^{6}}{7}+2\displaystyle\sum_{n\ge 1}^{}\left( \left( \dfrac{6}{n^{2}}\pi ^{4}-\dfrac{120}{n^{4}}\pi ^{2}+\dfrac{720 }{n^{6}}\right)\cos n\pi \right) \cos nx.\tag{1}$$
The computation is as follows:
$$\begin{equation*}
f(x)=x^{2p}=\frac{a_{0,2p}}{2}+\sum_{n=1}^{\infty }\left( a_{n,2p}\cos
nx+b_{n,2p}\sin nx\right) ,
\end{equation*}$$
where the coefficients are given by the following integrals
$$\begin{eqnarray*}
a_{0,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\;\mathrm{d}x=\frac{2\pi ^{2p}}{2p+1},
\\
a_{n,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\cos nx\;\mathrm{d}x=\frac{2}{\pi }
\int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x, \\
b_{n,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\sin nx\;\mathrm{d}x=0.
\end{eqnarray*}$$
The series expansion is thus
$$\begin{equation*}
x^{2p}=\frac{\pi ^{2p}}{2p+1}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos
nx\int_{0}^{\pi }t^{2p}\cos nt\;\mathrm{d}t.\tag{2}
\end{equation*}$$
For $f(\pi )=\pi ^{2p}$ we obtain
$$
\begin{equation*}
\pi ^{2p}=\frac{\pi ^{2p}}{2p+1}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi
\int_{0}^{\pi }t^{2p}\cos nt\;\mathrm{d}t,
\end{equation*}$$
where the integral
$$
\begin{equation*}
I_{n,2p}:=\int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x
\end{equation*}$$
satisfies the following recurrence, as can be shown by integration by parts
$$\begin{equation*}
I_{n,2p}=\frac{2p}{n^{2}}\pi ^{2p-1}\cos n\pi -\frac{2p(2p-1)}{n^{2}}
I_{n,2\left( p-1\right) },\qquad I_{n,0}=0.\tag{3}
\end{equation*}$$

*

*For $p=1$, we get
$$\begin{equation*}
I_{n,2}=\frac{2}{n^{2}}\pi\cos n\pi. 
\end{equation*}$$
and
$$\begin{eqnarray*}
\pi ^{2} &=&\frac{\pi ^{2}}{3}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi
\cdot I_{n,2} \\
&=&\frac{\pi ^{2}}{3}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi \left( 
\frac{2}{n^{2}}\pi \cos n\pi \right)  \\
&=&\frac{\pi ^{2}}{3}+4\sum_{n=1}^{\infty }\frac{1}{n^{2}} \\
&\Rightarrow &\zeta (2)=\sum_{n=1}^{\infty }\frac{1}{n^{2}}=\frac{\pi ^{2}}{6
}
\end{eqnarray*}$$

*For $p=2$, we get
$$
\begin{equation*}
I_{n,4}=\left( \frac{4\pi ^{3}}{n^{2}}-\frac{24\pi }{n^{4}}\right) \cos n\pi 
\end{equation*}$$
and
$$
\begin{eqnarray*}
\pi ^{4} &=&\frac{\pi ^{4}}{5}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi
\cdot I_{n,4}=\frac{\pi ^{4}}{5}+\frac{4\pi ^{4}}{3}-48\sum_{n=1}^{\infty }
\frac{1}{n^{4}} \\
&\Rightarrow &\zeta (4)=\sum_{n=1}^{\infty }\frac{1}{n^{4}}=\frac{\pi ^{4}}{
90}.
\end{eqnarray*}$$

*Finally for $p=3$, we get
$$\begin{equation*}
I_{n,6}=\left( \frac{6\pi ^{5}}{n^{2}}-\frac{120\pi ^{3}}{n^{4}}+\frac{720}{
n^{6}}\right) \cos n\pi 
\end{equation*}$$
and
$$
\begin{equation*}
\pi ^{6}=\frac{\pi ^{6}}{7}+2\sum_{n=1}^{\infty }\left( \frac{6\pi ^{4}}{
n^{2}}-\frac{120\pi ^{2}}{n^{4}}+\frac{720}{n^{6}}\right), 
\end{equation*}$$
from which the result follows
$$\zeta(6)=
\begin{equation*}
\sum_{n=1}^{\infty }\frac{1}{n^{6}}=\frac{\pi ^{6}}{945}.
\end{equation*}$$
Plots of the periodic function defined in $\left[ -\pi ,\pi \right] $ by $f(x)=x^{6}$ (blue curve) and of the partial sum with the first 10 terms of its Fourier trigonometric series (red curve).

This method generates recursively the sequence $(\zeta(2p))_{p\ge 1}$.
A: You may start with (for $x \in [0,1]$)
$$B_1(x)=x-\frac 12=2\sum_{n=1}^{\infty} \frac{(-1)^n}{2\pi n}\sin\left(2\pi n\left(x-\frac 12\right)\right)$$
and integrate multiple times $x-\frac 12$ (adding the necessary constant when required!) to get :
$$B_{2k}(x)=2(-1)^k\sum_{n=1}^{\infty} \frac{(-1)^n}{(2\pi n)^{2k}}\cos\left(2\pi n\left(x-\frac 12\right)\right)$$
(this is detailed in this paper page 6)
You could too use following formula
$$
\cot(\pi z)=\frac1{\pi}\left[\frac1{z}-\sum_{k=1}^{\infty}\frac{2z}{k^2-z^2}\right]
$$
(obtained from computation of Fourier series of $\cos(zx)$ for $-\pi \le x \le \pi$ with the result :
$$
\cos(zx)=\frac{2z\sin(\pi z)}{\pi}\left[\frac1{2z^2}+\frac{\cos(1x)}{1^2-z^2}-\frac{\cos(2x)}{2^2-z^2}+\frac{\cos(3x)}{3^2-z^2}-\cdots\right]
$$
applied to $x=\pi$)
and deduce an interesting expansion of $\frac z2\left(\cot\left(\frac z2\right)\right)$ in powers of $z^2$.
A: Similar to @RobJohn's solution: Setting $z=\pi$ in the Fourier series of $\cos(z x)$:
$$\cos(zx)=\frac{2x\sin(\pi x)}{\pi}\left[\frac1{2x^2}-\sum_{k=1}^\infty\frac{(-1)^k\cos(kz)}{k^2-x^2}\right],\quad x\notin\mathbb{Z}$$
gives
\begin{align}
\cot(\pi x)&=\frac1{\pi x}-\frac2{\pi x}\sum_{k=1}^\infty\frac{x^2}{k^2-x^2}\\
&=\frac1{\pi x}-\frac2{\pi x}\sum_{k=1}^\infty\frac{\frac{x^2}{k^2}}{1-\frac{x^2}{k^2}}\\
&=\frac1{\pi x}-\frac2{\pi x}\sum_{k=1}^\infty\sum_{n=1}^\infty\left(\frac{x^2}{k^2}\right)^n\\
&=\frac1{\pi x}-\frac2{\pi x}\sum_{n=1}^\infty x^{2n}\sum_{k=1}^\infty\frac{1}{k^{2n}}\\
&=\frac1{\pi x}-\frac2{\pi x}\color{red}{\sum_{n=1}^\infty \zeta(2n)x^{2n}}\\
\end{align}
Expanding $\cot x$ in Taylor series:
$$\cot x=\frac1{x}\sum_{n=0}^\infty\!\frac{(-1)^{n} B_{2n}}{(2n)!}(2x)^{2n}$$
or
\begin{align}
\cot(\pi x)&=\frac1{\pi x}\sum_{n=0}^\infty\!\frac{(-1)^{n} B_{2n}}{(2n)!}(2\pi x)^{2n}\\
&=\frac1{\pi x}-\frac2{\pi x}\color{red}{\sum_{n=1}^\infty\!\frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{2(2n)!}x^{2n}}
\end{align}
yields
$$\sum_{n=1}^\infty\!\zeta(2n)x^{2n}
=\sum_{n=1}^\infty\!\frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{2(2n)!}x^{2n}$$
Compare the coefficients of $x^{2n}$ to get
$$\zeta(2n)=\frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{2(2n)!}.$$
A: Let $$g(x) = \sum_{n=1}^{\infty} \frac{\cos(2nx)}{n^2}$$
Then
$$g^3(x)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{k=1}^{\infty} \frac{\cos(2nx)\cos(2mx)\cos(2kx)}{n^2m^2k^2}$$
Since $\cos(a)\cos(b)=\frac{1}{2}\left(\cos(a+b)+\cos(a-b)\right)$ then
$$g^3(x)=\frac{1}{2}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{k=1}^{\infty} \frac{\cos(2nx)\cos(2mx+2kx)+\cos(2nx)\cos(2mx-2kx)}{n^2m^2k^2}$$
Due to orthogonality of cosine function we get after integration : 
$$\int_0^\pi g^3(x)\;\mathrm{d}x=\frac{\pi}{4}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{k=1}^{\infty} \frac{\delta_{n,m+k}+\delta_{n,m-k}}{n^2m^2k^2}=
\frac{\pi}{2}\sum_{m=1}^{\infty}\sum_{k=1}^{\infty} \frac{1}{(m+k)^2m^2k^2}$$
