Riemann Xi function

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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[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]

\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[edit]

The general form for even integers is

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

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

\xi(2) = {\pi \over 6}

Series representations[edit]

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

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

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

To ensure convergence in the latter formula, 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 combined.

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.