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, $$\xi$$ was renamed with an upper-case $$~\Xi~$$ (Greek letter "Xi") by Edmund Landau. Landau's lower-case $$~\xi~$$ ("xi") is defined as
 * $$\xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$$

for $$s \in \mathbb{C}$$. 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
 * $$\xi(1-s) = \xi(s)~.$$

Riemann's original function, rebaptised upper-case $$~\Xi~$$ by Landau, satisfies
 * $$\Xi(z) = \xi \left(\tfrac{1}{2} + z i \right)$$,

and obeys the functional equation
 * $$\Xi(-z) = \Xi(z)~.$$

Both functions are entire and purely real for real arguments.

Values
The general form for positive even integers is


 * $$\xi(2n) = (-1)^{n+1}\frac{n!}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n-1)$$

where Bn denotes the n-th Bernoulli number. For example:


 * $$\xi(2) = {\frac{\pi}{6}} $$

Series representations
The $$\xi$$ function has the series expansion


 * $$\frac{d}{dz} \ln \xi \left(\frac{-z}{1-z}\right) =

\sum_{n=0}^\infty \lambda_{n+1} z^n,$$

where


 * $$\lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n}

\left[s^{n-1} \log \xi(s) \right] \right|_{s=1} = \sum_{\rho} \left[1- \left(1-\frac{1}{\rho}\right)^n\right],$$

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 &gt; 0 for all positive n.

Hadamard product
A simple infinite product expansion is


 * $$\xi(s) = \frac12 \prod_\rho \left(1 - \frac{s}{\rho} \right),\!$$

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.