Riemann xi function

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Riemann xi function
x
(
s
)
{\displaystyle \xi (s)}
in the complex plane. The color of a point
s
{\displaystyle s}
encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument. Riemann Xi cplot.svg
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.

Contents

Definition

Riemann's original lower-case "xi"-function, was renamed with a (Greek uppercase letter "xi") by Edmund Landau. Landau's (lower-case "xi") is 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, renamed as the upper-case by Landau, [1] satisfies

and obeys the functional equation

Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is

where denotes the th Bernoulli number. For example:

Series representations

The function has the series expansion

where

where the sum extends over , 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 for all positive .

Hadamard product

A simple infinite product expansion is

where ranges over the roots of .

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 and should be grouped together.

References

  1. 1 2 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. §70-71 and page 894.

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