# Riemann Xi function

Riemann xi function $\xi(s)$ in the complex plane. The color of a point $s$ encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument.

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.

## Definition

Riemann's original lower-case xi-function, ξ, has been renamed with an upper-case Xi, Ξ, by Edmund Landau (see below). Landau's lower-case xi, ξ, is defined as:[1]

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

for $s\in\Bbb{C}$. Here ζ(s) denotes the Riemann zeta function and Γ(s) is the Gamma function. The functional equation (or reflection formula) for xi is

$\xi(1-s) = \xi(s).$

The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as

$\Xi(z) = \xi(\frac12+zi)$

and obeys the functional equation

$\Xi(-z) =\Xi(z).$

As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ.

## Values

The general form for even integers is

$\xi(2n) = (-1)^{n+1}\frac{1}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n^2-n)(n-1)!$

where Bn denotes the n-th Bernoulli number. For example:

$\xi(2) = {\pi \over 6}$

## Series representations

The xi function has the series expansion

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

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

A simple infinite product expansion is

$\Xi(s) = \Xi(0)\prod_\rho \left(1 - \frac{s}{\rho} \right).\!$

To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be combined.

## References

1. ^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.