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, was renamed with an upper-case (Greek letter "Xi") by Edmund Landau. Landau's lower-case ("xi") which Landau defined as:
for . Here denotes the Riemann zeta function and is the Gamma function. The functional equation (or reflection formula) for Landau's is
Riemann's original function is re-defined by Landau as upper-case ::§71
and obeys the functional equation
Landau reports:894 that the function above is the function Riemann originally denoted by .
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.
- ^ a b c Landau, Edmund (1974) . Handbuch der Lehre von der Verteilung der Primzahlen [Handbook of the Study of Distribution of the Prime Numbers] (Third ed.). New York: Chelsea. §70.
This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.