value of the Riemann zeta function at s=0


Let ζ denote the meromorphic extension of the Riemann zeta functionDlmfDlmfMathworldPlanetmath to the complex plane. Then ζ(0)=-12.


Recall that one of the for the Riemann zeta function in the critical stripMathworldPlanetmath is given by


where [x] denotes the integer part of x.

Also recall the functional equation


where Γ denotes the gamma functionDlmfDlmfMathworldPlanetmath.

The only pole ( of ζ occurs at s=1. Therefore, ζ is analytic, and thus continuous, at s=0.

Let lims0+ denote the limit as s approaches 0 along any path contained in the region Re(s)>0. Thus:

ζ(0) =lims0+ζ(s)

Title value of the Riemann zeta function at s=0
Canonical name ValueOfTheRiemannZetaFunctionAtS0
Date of creation 2013-03-22 16:07:17
Last modified on 2013-03-22 16:07:17
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 20
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 11M06
Related topic CriticalStrip
Related topic FormulaeForZetaInTheCriticalStrip