Riemann Xi function
From Wikipedia, the free encyclopedia
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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[edit] Definition
Riemann's lower case xi is defined as:
The functional equation (or reflection formula) for the xi is
- ξ(1 − s) = ξ(s).
[edit] Values
The general form for even integers is
For example:
[edit] Series representations
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 λn > 0 for all positive n.
[edit] References
- Eric W. Weisstein, Xi-Function at MathWorld.
This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the GFDL.
