Riemann xi function

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title: "Riemann xi function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["zeta-and-l-functions", "bernhard-riemann"] description: "Simpler variant of the Riemann zeta function" topic_path: "general/zeta-and-l-functions" source: "https://en.wikipedia.org/wiki/Riemann_xi_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Simpler variant of the Riemann zeta function ::

[[Image:Riemann Xi cplot.svg|right|thumb|300px|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 [[complex number|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

Riemann's original lower-case "xi"-function, \xi was renamed with a \Xi (Greek uppercase letter "xi") by Edmund Landau. Landau's \xi (lower-case "xi") is defined as : \xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s) for s \in \mathbb{C}. Here \zeta(s) denotes the Riemann zeta function and \Gamma(s) is the gamma function.

The functional equation (or reflection formula) for Landau's \xi is : \xi(1-s) = \xi(s) . Riemann's original function, renamed as the upper-case \Xi by Landau, satisfies : \Xi(z) = \xi \left(\tfrac{1}{2} + z i \right) , and obeys the functional equation : \Xi(-z) = \Xi(z) . Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is : \xi(2n) = (-1)^{n+1}\frac{n!}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n-1) where B_n denotes the th Bernoulli number. For example: : \xi(2) = {\frac{\pi}{6}}

Series representations

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, where : \lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n} \left[s^{n-1} \log \xi(s) \right] \right|{s=1} = \sum{\rho} \left[ 1- \left(1-\frac{1}{\rho}\right)^n \right], where the sum extends over \rho, the non-trivial zeros of the zeta function, in order of \vert\Im(\rho)\vert.

This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having \lambda_n 0 for all positive n.

Hadamard product

A simple infinite product expansion is : \xi(s) = \frac12 \prod_\rho \left(1 - \frac{s}{\rho} \right), where \rho ranges over the roots of \xi.

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 \rho and \bar\rho should be grouped together.

References

• {{cite journal |first1=J.B. |last1=Keiper |journal=Mathematics of Computation |year=1992 |volume=58 |issue=198 |pages=765–773 |title=Power series expansions of Riemann's xi function |doi=10.1090/S0025-5718-1992-1122072-5 |doi-access=free |bibcode=1992MaCom..58..765K

References

1. Landau, Edmund. (1974). "Handbuch der Lehre von der Verteilung der Primzahlen". Chelsea.

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