# Fourier Series and the Riemann-Zeta Function

1. Apr 9, 2007

### MaGG

1. The problem statement, all variables and given/known data
Use the Fourier series technique to show that the following series sum to the quantities shown:
$$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$$
$$1+\frac{1}{3^2}+\frac{1}{5^2}+...=\frac{\pi^2}{8}$$

2. Relevant equations
I know the Riemann-Zeta function is
$$\zeta (m)=\sum_{n=1}^\infty \frac{1}{n^m}$$

3. The attempt at a solution
For the first series, I know that it's the evaluation of $\zeta (4)$. The problem is getting the derivation down. I've found an example of it being solved and proven at $\zeta (2)$ here: http://planetmath.org/?op=getobj&from=objects&name=ValueOfTheRiemannZetaFunctionAtS2 . The problem is, I don't really understand what to define as the initial function $f(x)$.

For the second series, I think I've found it to be:
$$\frac{1}{(2n-1)^2}$$
After that, I don't know how to go about solving it using the Fourier method. Any help would be greatly appreciated, thanks!

2. Apr 9, 2007

### e(ho0n3

By "Fourier series technique", do you mean expand the 1/n4 or the 1/(2n - 1)2 into its Fourier series?

3. Sep 25, 2008