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, was renamed with an upper-case (Greek letter "Xi") by Edmund Landau. Landau's lower-case ("xi") which Landau defined as:[1]

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 :[1]:§71

and obeys the functional equation

Landau reports[1]:894 that the function above is the function Riemann originally denoted by . Both functions are entire and purely real for real arguments.

Values[edit]

The general form for positive 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. ^ a b c Landau, Edmund (1974) [1909]. Handbuch der Lehre von der Verteilung der Primzahlen [Handbook of the Study of Distribution of the Prime Numbers] (Third ed.). New York: Chelsea. §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.