Riemann xi function
in the complex plane
. The color of a point
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.
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:
for . Here ζ(s) denotes the Riemann zeta function and Γ(s) is the Gamma function. The functional equation (or reflection formula) for xi is
The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as
and obeys the functional equation
As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ.
Both functions are entire and purely real for real arguments.
The general form for positive even integers is
where Bn denotes the n-th Bernoulli number. For example:
The function has the series expansion
where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of .
This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λn > 0 for all positive n.
A simple infinite product expansion is
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.
- ^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.
This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.