Riemann xi function

The function is named in honour of Bernhard Riemann.

(Greek letter "Xi") by Edmund Landau.

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.

The general form for positive even integers is where Bn denotes the n-th Bernoulli number.

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. 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.