functional equation for the Riemann Xi function

The Riemann Xi Function satisfies the following functional equation:


This equation directly implies the Riemann Zeta functionDlmfDlmfMathworldPlanetmath’s functional equation.

This equation plays an important role in the theory of the Riemann Zeta function. It allows one to analytically continue the Zeta and the Xi functionsMathworldPlanetmath to the whole complex planeMathworldPlanetmath. The definition of the Zeta functionMathworldPlanetmath as a series is only valid when (s)>1. By using this equation, one can express the values of these two functions when (s)<1 in terms of the values when (s)>1. As an illustration of its importance, one can cite the fact that there are no zeros of the Zeta function with real partDlmfMathworld greater than 1, so without this functional equation the study of the Zeta function would be very limited.

Title functional equation for the Riemann Xi function
Canonical name FunctionalEquationForTheRiemannXiFunction
Date of creation 2013-03-22 13:24:15
Last modified on 2013-03-22 13:24:15
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Theorem
Classification msc 11M06