# Riemann $\Xi$ function

The Riemann Xi function

 $\Xi(s)=\pi^{-\frac{1}{2}s}\Gamma(\frac{1}{2}s)\zeta(s),$

(where $\Gamma(s)$ is Euler’s Gamma function and $\zeta(s)$ is the Riemann zeta function), is the key to the functional equation for the Riemann zeta function.

Riemann himself used the notation of a lower case xi ($\xi$). The famous Riemann hypothesis is equivalent to the assertion that all the zeros of $\xi$ are real, in fact Riemann himself presented his original hypothesis in terms of that function.

Riemann’s lower case xi is defined as

 $\xi(s)=\frac{1}{2}s(s-1)\Xi(s).$
Title Riemann $\Xi$ function RiemannXiFunction 2013-03-22 13:24:06 2013-03-22 13:24:06 PrimeFan (13766) PrimeFan (13766) 11 PrimeFan (13766) Definition msc 11M06