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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.

Contents

Definition

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

\xi(1-s)=\xi(s).\,

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.

Values

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.

References

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


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