Riemann $\mathrm{\Xi }$ function

The Riemann Xi function

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

(where $\mathrm{\Gamma }(s)$ 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

$$\xi (s)=\frac{1}{2}s(s-1)\mathrm{\Xi }(s).$$

Title	Riemann $\mathrm{\Xi }$ function
Canonical name	RiemannXiFunction
Date of creation	2013-03-22 13:24:06
Last modified on	2013-03-22 13:24:06
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	11
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11M06