The Full Wiki

More info on Riemann Xi function

Riemann Xi function: Wikis


Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.


From Wikipedia, the free encyclopedia

Riemann xi function ξ(s) in the complex plane. The color of a point s encodes the value of the function. Strong colors denote values close to zero and hue encodes the value's argument.

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.



Riemann's lower-case xi is defined as:

 \xi(s) = \frac{1}{2} s(s-1) \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s).

The functional equation (or reflection formula) for the xi is


The upper-case Xi function is defined as

 \Xi(s) = \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s)

and of course obeys the same functional equation.


The general form for even integers is

\xi(2n) = (-1)^{n+1}{{B_{2n}2^{2n-1}\pi^{n}(2n^2-n)(n-1)!} \over {(2n)!}}.

For example:

\xi(2) = {\pi \over 6}.

Series representations

The xi function has the series expansion

\frac{d}{dz} \log \xi \left(\frac{-z}{1-z}\right) = \sum_{n=0}^\infty \lambda_{n+1} z^n.

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.


This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


Got something to say? Make a comment.
Your name
Your email address