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

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

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

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

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

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

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

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

Further 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.