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 an upper-case \( {\displaystyle ~\Xi ~} \) (Greek letter "Xi") by Edmund Landau. Landau's lower-case \( {\displaystyle ~\xi ~} \) ("xi") is defined as[1]
\( {\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)} \)
for \( {\displaystyle s\in \mathbb {C} } \) . Here \( \zeta (s) \) denotes the Riemann zeta function and \( \Gamma(s) \) is the Gamma function. The functional equation (or reflection formula) for Landau's \( {\displaystyle ~\xi ~} \) is
\( {\displaystyle \xi (1-s)=\xi (s)~.}
Riemann's original function, rebaptised upper-case \( {\displaystyle ~\Xi ~} \) by Landau,[1] satisfies
\( {\displaystyle \Xi (z)=\xi \left({\tfrac {1}{2}}+zi\right)}, \)
and obeys the functional equation
\( {\displaystyle \Xi (-z)=\Xi (z)~.} \)
Both functions are entire and purely real for real arguments.
Values
The general form for positive even integers is
\( {\displaystyle \xi (2n)=(-1)^{n+1}{\frac {n!}{(2n)!}}B_{2n}2^{2n-1}\pi ^{n}(2n-1)} \)
where Bn denotes the n-th Bernoulli number. For example:
\( {\displaystyle \xi (2)={\frac {\pi }{6}}} \)
Series representations
The ξ {\displaystyle \xi } \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 ρ, the non-trivial zeros of the zeta function, in order of \( |\Im (\rho )|. \)
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
A simple infinite product expansion is
\( {\displaystyle \xi (s)={\frac {1}{2}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right),\!} \)
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
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.
Further references
Weisstein, Eric W. "Xi-Function". MathWorld.
Keiper, J.B. (1992). "Power series expansions of Riemann's xi function". Mathematics of Computation. 58 (198): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.
This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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