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Riemann xi function
From Wikipedia, the free encyclopedia
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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.
This article relies largely or entirely on a single source. (September 2025) |

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Definition
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Riemann's original lower-case "xi"-function, was renamed with a (Greek uppercase letter "xi") by Edmund Landau. Landau's (lower-case "xi") is defined as[1]
for . Here denotes the Riemann zeta function and is the gamma function.
The functional equation (or reflection formula) for Landau's is
Riemann's original function, renamed as the upper-case by Landau,[1] satisfies
and obeys the functional equation
Both functions are entire and purely real for real arguments.
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Values
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The general form for positive even integers is
where denotes the th Bernoulli number. For example:
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Series representations
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The function has the series expansion
where
where the sum extends over , the non-trivial zeros of the zeta function, in order of .
This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having for all positive .
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Hadamard product
A simple infinite product expansion is
where ranges over the roots of .
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 and should be grouped together.
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References
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