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.
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Riemann's lowercase xi is defined as:
The functional equation (or reflection formula) for the xi is
The uppercase Xi function is defined as
and of course obeys the same functional equation.
The general form for even integers is
For example:
The xi function has the series expansion
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.
This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
