# value of the Riemann zeta function at $s=2$

Here we present an application of Parseval’s equality to number theory. Let $\zeta(s)$ denote the Riemann zeta function. We will compute the value

 $\zeta(2)$

with the help of Fourier analysis.

Example:

Let $f\colon\mathbb{R}\to\mathbb{R}$ be the “identity” function, defined by

 $f(x)=x,\text{ for all }x\in\mathbb{R}.$

The Fourier series of this function has been computed in the entry example of Fourier series.

Thus

 $\displaystyle f(x)=\ x$ $\displaystyle=$ $\displaystyle a_{0}^{f}+\sum_{n=1}^{\infty}(a_{n}^{f}\cos(nx)+b_{n}^{f}\sin(nx))$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{n}\sin(nx),\quad\forall x% \in(-\pi,\pi).$

Parseval’s theorem asserts that:

 $\frac{1}{\pi}\int_{-\pi}^{\pi}f^{2}(x)dx=2(a_{0}^{f})^{2}+\sum_{k=1}^{\infty}[% (a_{k}^{f})^{2}+(b_{k}^{f})^{2}].$

So we apply this to the function $f(x)=x$:

 $2(a_{0}^{f})^{2}+\sum_{k=1}^{\infty}[(a_{k}^{f})^{2}+(b_{k}^{f})^{2}]=\sum_{n=% 1}^{\infty}\frac{4}{n^{2}}=4\sum_{n=1}^{\infty}\frac{1}{n^{2}}$

and

 $\frac{1}{\pi}\int_{-\pi}^{\pi}f^{2}(x)dx=\frac{1}{\pi}\int_{-\pi}^{\pi}x^{2}dx% =\frac{2\pi^{2}}{3}.$

Hence by Parseval’s equality

 $4\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{2\pi^{2}}{3}$

and hence

 $\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}.$
Title value of the Riemann zeta function at $s=2$ ValueOfTheRiemannZetaFunctionAtS2 2013-03-22 13:57:16 2013-03-22 13:57:16 alozano (2414) alozano (2414) 15 alozano (2414) Theorem msc 11M99 msc 42A16 ExampleOfFourierSeries PersevalEquality ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers ValueOfRiemannZetaFunctionAtS4 ValueOfDirichletEtaFunctionAtS2 APathologicalFunctionOfRiemann KummersAccelerationMethod