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| [[Image:Complex Riemann Xi.jpg|right|thumb|300px|Riemann xi function <math> \xi(s) </math> in the [[complex plane]]. The color of a point <math> s </math> encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's [[complex number|argument]].]]
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| 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]].
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| ==Definition==
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| 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:<ref>[[Edmund Landau]]. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.</ref>
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| :<math>\xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s)</math>
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| for <math>s\in\Bbb{C}</math>. Here ζ(''s'') denotes the [[Riemann zeta function]] and Γ(s) is the [[Gamma function]]. The functional equation (or [[reflection formula]]) for xi is
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| :<math>\xi(1-s) = \xi(s).</math> | |
| The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as
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| :<math>\Xi(z) = \xi(\frac12+zi)</math>
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| and obeys the functional equation
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| :<math>\Xi(-z) =\Xi(z).</math>
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| As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ.
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| ==Values==
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| The general form for even integers is
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| :<math>\xi(2n) = (-1)^{n+1}\frac{1}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n^2-n)(n-1)!</math>
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| where ''B<sub>n</sub>'' denotes the ''n''-th [[Bernoulli number]]. For example:
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| :<math>\xi(2) = {\pi \over 6} </math>
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| ==Series representations==
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| The xi function has the series expansion
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| :<math>\frac{d}{dz} \ln \xi \left(\frac{-z}{1-z}\right) =
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| \sum_{n=0}^\infty \lambda_{n+1} z^n</math>
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| This expansion plays a particularly important role in [[Li's criterion]], which states that the [[Riemann hypothesis]] is equivalent to having λ<sub>''n''</sub> > 0 for all positive ''n''.
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| ==Hadamard product==
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| A simple [[infinite product]] expansion is
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| :<math>\Xi(s) = \frac{1}{2}\prod_\rho \left(1 - \frac{s}{\rho} \right).\!</math>
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| 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.
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| ==References==
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| {{reflist}}
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| ==Further references==
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| * {{mathworld|urlname=Xi-Function|title=Xi-Function}}
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| * {{cite journal
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| |first1=J.B.
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| |last1=Keiper
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| |journal=Mathematics of Computation
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| |year=1992
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| |volume=58
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| |issue=198
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| |pages=765–773
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| |title=Power series expansions of Riemann's xi function
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| |doi=10.1090/S0025-5718-1992-1122072-5
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| |bibcode=1992MaCom..58..765K
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| }}
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| {{PlanetMath attribution|id=3943|title=Riemann Ξ function}}
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| [[Category:Zeta and L-functions]]
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