Riemann Xi function
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, was renamed with an upper-case (Greek 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, rebaptised 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 Bn denotes the n-th Bernoulli number. For example:
Series representations
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
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
- ^ a b 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.
- 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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