Riemann Xi function

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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.

Definition[edit]

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]

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 would be entire functions if you filled every removable singularity in.

Values[edit]

The general form for even integers is

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

Series representations[edit]

The function has the series expansion

where

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.

Hadamard product[edit]

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.

References[edit]

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

Further references[edit]

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