value of the Riemann zeta function at
Here we present an application of Parseval’s equality to number
theory. Let denote the Riemann zeta function
. We will
compute the value
with the help of Fourier analysis.
Example:
Let be the “identity” function,
defined by
The Fourier series of this function has been computed in the entry
example of Fourier series.
Thus
Parseval’s theorem asserts that:
So we apply this to the function :
and
Hence by Parseval’s equality
and hence
Title | value of the Riemann zeta function at |
---|---|
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 |
Related topic | KummersAccelerationMethod |