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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.
Contents
Definition
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:
for
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. Both would be entire functions if you filled every removable singularity in.
Values
The general form for even integers is
where Bn denotes the n-th Bernoulli number. For example:
Series representations
The
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.