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Riemann xi function

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Riemann xi function
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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.

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Riemann xi function in the complex plane. The color of a point encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument.
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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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