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 $f:ℝ\to ℝ$ be the “identity” function, defined by

$$f(x)=x,\text{ for all }x\in ℝ.$$

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

Thus

$f(x)=x$ $=$ $a_0^f+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(a_n^f\cos (nx)+b_n^f\sin (nx))$

$=$ ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{(-1)}^{n+1}{\displaystyle \frac{2}{n}}\sin (nx),\forall x\in (-\pi ,\pi ).$

Parseval’s theorem asserts that:

$$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{f}^2(x)𝑑x=2{(a_0^f)}^2+\sum _{k=1}^{\mathrm{\infty }}[{(a_k^f)}^2+{(b_k^f)}^2].$$

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

$$2{(a_0^f)}^2+\sum _{k=1}^{\mathrm{\infty }}[{(a_k^f)}^2+{(b_k^f)}^2]=\sum _{n=1}^{\mathrm{\infty }}\frac{4}{n^2}=4\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$$

and

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

Hence by Parseval’s equality

$$4\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{2{\pi }^2}{3}$$

and hence

$$\zeta (2)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}.$$

Title	value of the Riemann zeta function at $s=2$
Canonical name	ValueOfTheRiemannZetaFunctionAtS2
Date of creation	2013-03-22 13:57:16
Last modified on	2013-03-22 13:57:16
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	15
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11M99
Classification	msc 42A16
Related topic	ExampleOfFourierSeries
Related topic	PersevalEquality
Related topic	ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers
Related topic	ValueOfRiemannZetaFunctionAtS4
Related topic	ValueOfDirichletEtaFunctionAtS2
Related topic	APathologicalFunctionOfRiemann
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