Riemann xi function

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.

Riemann's original lower-case "xi"-function, ${\displaystyle \xi }$ was renamed with a ${\displaystyle \Xi }$ (Greek uppercase letter "xi") by Edmund Landau. Landau's ${\displaystyle \xi }$ (lower-case "xi") is defined as¹

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

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, renamed as the 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.

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 ${\displaystyle B_{n}}$ denotes the ⁠${\displaystyle n}$⁠th Bernoulli number. For example:

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

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 ${\displaystyle \rho }$, the non-trivial zeros of the zeta function, in order of ${\displaystyle \vert \Im (\rho )\vert }$.

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

A simple infinite product expansion is

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

where ${\displaystyle \rho }$ ranges over the roots of ${\displaystyle \xi }$.

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 ${\displaystyle \rho }$ and ${\displaystyle {\bar {\rho }}}$ should be grouped together.