Riemann Ξ 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
:$\xi(s) = \frac s(s-1) \pi^ \Gamma\left(\frac\right) \zeta(s)$
for $s \in \mathbb$. 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
:$\Xi(z) = \xi \left(\tfrac + z i \right)$,
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

:$\xi(2n) = (-1)^\fracB_2^\pi^(2n-1)$

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

:$\xi(2) =$

Series representations

The $\xi$ function has the series expansion

:$\frac \ln \xi \left(\frac\right) = \sum_^\infty \lambda_ z^n,$

where

:\lambda_n = \frac \left. \frac
\left ^ \log \xi(s) \right\_ = \sum_ \left -
\left(1-\frac\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 λ_''n'' > 0 for all positive ''n''.

Hadamard product

A simple infinite product expansion is

:$\xi(s) = \frac12 \prod_\rho \left(1 - \frac \right),\!$
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.

References

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{{Bernhard Riemann

Zeta and L-functions
Bernhard Riemann