Riemann Xi function

From Wikipedia, the free encyclopedia

In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.

[edit] Definition

Riemann's lower case xi is defined as:

$\xi(s) = \frac{1}{2} s(s-1) \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s).$

The functional equation (or reflection formula) for the xi is

ξ(1 - s) = ξ(s).

[edit] Series representations

The xi function has the series expansion

$\frac{d}{dz} \log \xi \left(\frac{-z}{1-z}\right) = \sum_{n=0}^\infty \lambda_{n+1} z^n.$

This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having $λ n > 0$ for all positive n.