Riemann Xi function

Riemann xi function $\xi(s)$ 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.

Definition

Riemann's lower-case xi, ξ, 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, Ξ, 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

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

For example:

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

Series representations

The xi function has the series expansion

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

This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having $\lambda_n >0$ for all positive n.

Riemann Hypothesis

(This contribution is interesting yet completely unintelligible. The operators D, N, H and z have not been defined, neither has the functional space over which the determinant operates. Even if the reader is interested, no source material is provided to support these "claims" (technically, nothing has been claimed until the symbols have meaning, which they haven't.))

As pointed by several works by Alain Connes and others, the Riemann hypothesis is equivalent to the assertion that the Riemann xi function is the functional determinant of the operator

$-D^{2}+f(x) \,$

with

$f^{-1}(x) = \sqrt {4\pi} \frac{d^{\frac{1}{2}}N(x)}{dx^{\frac{1}{2}}}, \text{ so } \frac{\xi\left(\frac{1}{2}+iz\right)}{\xi\left(\frac{1}{2}\right)} = \frac{\det(H-z^{2})}{\det(H)}$

this conjecture is supported by several numerical evaluations

References

• Weisstein, Eric W., "Xi-Function" from MathWorld.
• Keiper, J.B. (1992). "Power series expansions of Riemann's xi function". Mathematics of Computation 58 (198): 765–773. Bibcode 1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5. 

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