value of Dirichlet eta function at s=2

The value


of the Dirichlet eta functionMathworldPlanetmath can be found by using the Fourier cosine seriesMathworldPlanetmath of the functionMathworldPlanetmathxx-x2 on the interval[0, 1]:

x-x2=16-1π2n=1cos2nπxn2for  0x1 (1)

Substituting  x:=12  to the equation (1) yields


which we can solve to the form

η(2)=n=1(-1)n+1n2=π212. (2)

This result could be obtained very simply by using the functional equation connecting Dirichlet eta function to Riemann zeta functionDlmfDlmfMathworldPlanetmath.

Combining the equation (2) with the result concerning the Riemann zeta function at 2 ( shows that

1+132+152+172+=π28. (3)
Title value of Dirichlet eta function at s=2
Canonical name ValueOfDirichletEtaFunctionAtS2
Date of creation 2013-03-22 18:22:09
Last modified on 2013-03-22 18:22:09
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Result
Classification msc 11M41
Related topic CosineAtMultiplesOfStraightAngle
Related topic ValueOfTheRiemannZetaFunctionAtS2