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

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

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

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

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

Riemann's original function, rebaptised upper-case $\mathrm{\Xi }$ by Landau, satisfies

$\mathrm{\Xi }(z)=\xi (\frac{1}{2}+zi)$,

and obeys the functional equation

$\mathrm{\Xi }(-z)=\mathrm{\Xi }(z).$

Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is

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

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

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

Series representations

The $\xi$ function has the series expansion

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

where

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

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

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

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

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.

References

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