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.
The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as
and obeys the functional equation
As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ.
The general form for even integers is
where Bn denotes the n-th Bernoulli number. For example:
The function has the series expansion
where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of .
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.
- Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.
- 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.