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 original lower-case "xi"-function,
\xi
\Xi
\xi
\xi(s)=
| 1 | |
| 2 |
s(s-1)\pi-s/2\Gamma\left(
| s | |
| 2 |
\right)\zeta(s)
s\inC
\zeta(s)
\Gamma(s)
The functional equation (or reflection formula) for Landau's
\xi
\xi(1-s)=\xi(s).
\Xi
\Xi(z)=\xi\left(\tfrac{1}{2}+zi\right),
\Xi(-z)=\Xi(z).
The general form for positive even integers is
\xi(2n)=(-1)n+1
| n! | |
| (2n)! |
B2n22n-1\pin(2n-1)
Bn
\xi(2)={
| \pi | |
| 6 |
The
\xi
| d | |
| dz |
ln\xi\left(
| -z | |
| 1-z |
\right)=
| infty | |
| \sum | |
| n=0 |
λn+1zn,
λn=
| 1 | |
| (n-1)! |
\left.
| dn | |
| dsn |
\left[sn-1log\xi(s)\right]\right|s=1=\sum\rho\left[1-\left(1-
| 1 | |
| \rho |
\right)n\right],
\rho
\vert\Im(\rho)\vert
This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having
λn>0
n
A simple infinite product expansion is
\xi(s)=
| 12 | |
| \prod |
\rho\left(1-
| s | |
| \rho |
\right),
\rho
\xi
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
\rho
\bar\rho
