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 \Xi (Greek letter "Xi") by Edmund Landau. Landau's lower-case \xi ("xi") is defined as for. Here \zeta(s) denotes the Riemann zeta function and \Gamma(s) is the Gamma function. The functional equation (or reflection formula) for Landau's \xi is Riemann's original function, rebaptised upper-case \Xi by Landau, 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 \xi function has the series expansion where 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 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.