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 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.

Definition

Riemann's original lower-case "xi"-function, $\xi$ was renamed with an upper-case $~\Xi ~$ (Greek letter "Xi") by Edmund Landau. Landau's lower-case $~\xi ~$ ("xi") is defined as

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

for $s\in \mathbb {C}$. Here $\zeta (s)$ denotes the Riemann zeta function and $\Gamma (s)$ 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 $~\Xi ~$ by Landau, satisfies

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

and obeys the functional equation

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

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

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

where

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

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

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

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

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.

References

• Weisstein, Eric W. "Xi-Function". MathWorld.
• Keiper, J.B. (1992). "Power series expansions of Riemann's xi function". Mathematics of Computation. 58 (198): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.

This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.