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, ${\displaystyle \xi }$ was renamed with an upper-case ${\displaystyle ~\Xi ~}$ (Greek letter "Xi") by Edmund Landau. Landau's lower-case ${\displaystyle ~\xi ~}$ ("xi") is defined as^[1]

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

for ${\displaystyle s\in \mathbb {C} }$. Here ${\displaystyle \zeta (s)}$ denotes the Riemann zeta function and ${\displaystyle \Gamma (s)}$ is the Gamma function.

The functional equation (or reflection formula) for Landau's ${\displaystyle ~\xi ~}$ is

${\displaystyle \xi (1-s)=\xi (s)~.}$

Riemann's original function, rebaptised upper-case ${\displaystyle ~\Xi ~}$ by Landau,^[1] satisfies

${\displaystyle \Xi (z)=\xi \left({\tfrac {1}{2))+zi\right)}$,

and obeys the functional equation

${\displaystyle \Xi (-z)=\Xi (z)~.}$

Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is

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

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

${\displaystyle \xi (2)={\frac {\pi }{6))}$

Series representations

The ${\displaystyle \xi }$ function has the series expansion

${\displaystyle {\frac {d}{dz))\ln \xi \left({\frac {-z}{1-z))\right)=\sum _{n=0}^{\infty }\lambda _{n+1}z^{n},}$

where

${\displaystyle \lambda _{n}={\frac {1}{(n-1)!))\left.{\frac {d^{n)){ds^{n))}\left[s^{n-1}\log \xi (s)\right]\right|_{s=1}=\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho ))\right)^{n}\right],}$

where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of ${\displaystyle |\Im (\rho )|}$.

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.

Hadamard product

A simple infinite product expansion is

${\displaystyle \xi (s)={\frac {1}{2))\prod _{\rho }\left(1-{\frac {s}{\rho ))\right),\!}$

where ρ ranges over the roots of ξ.

To ensure convergence in the expansion, 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 grouped together.