Riemann Xi 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 lower case xi is defined as:

:$xi(s)\; =\; frac\{1\}\{2\}\; s(s-1)\; pi^\{-frac\{s\}\{2\; Gammaleft(frac\{s\}\{2\}\; ight)\; zeta(s).$

The functional equation (or reflection formula) for the xi is

:$xi(1-s)=xi(s).$

The upper case Xi function is defined as:$Xi(s)\; =\; pi^\{-frac\{s\}\{2\; Gammaleft(frac\{s\}\{2\}\; ight)\; zeta(s)$and of course obeys the same functional equation.

Values

The general form for even integers is

:$xi(2n)\; =\; (-1)^\{n+1\}$B_{2n}2^{2n-1}pi^{n}(2n^2-n)(n-1)!} over {(2n)!.

For example:

:$xi(2)\; =\; \{pi\; over\; 6\}.$

eries representations

The xi function has the series expansion

:$frac\{d\}\{dz\}\; log\; xi\; left(frac\{-z\}\{1-z\}\; ight)\; =\; sum\_\{n=0\}^infty\; lambda\_\{n+1\}\; z^n.$

This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having $lambda\_n\; >0$ for all positive "n".

References

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