Riemann Xi function

Riemann xi function

\xi (s)

in the complex plane. The color of a point

s

encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument.

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 original lower-case xi-function, ξ, has been renamed with an upper-case Xi, Ξ, by Edmund Landau (see below). Landau's lower-case xi, ξ, is defined as:^[1]

\xi (s)={\tfrac {1}{2}}s(s-1)\pi ^{{-s/2}}\Gamma \left({\tfrac {1}{2}}s\right)\zeta (s)

for $s\in {\mathbb {C}}$ . Here ζ(s) denotes the Riemann zeta function and Γ(s) is the Gamma function. The functional equation (or reflection formula) for xi is

\xi (1-s)=\xi (s).

The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as

\Xi (z)=\xi ({\frac 12}+zi)

and obeys the functional equation

\Xi (-z)=\Xi (z).

As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ. Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is

\xi (2n)=(-1)^{n+1}{\frac {n!}{(2n)!}}B_{2n}2^{2n-1}\pi ^{n}(2n-1)

where B_n denotes the n-th Bernoulli number. For example:

\xi (2)={\frac {\pi }{6}}

Series representations

The $\xi$ function has the series expansion

{\frac {d}{dz}}\ln \xi \left({\frac {-z}{1-z}}\right)=\sum _{{n=0}}^{\infty }\lambda _{{n+1}}z^{n},

where

\lambda _{n}={\frac {1}{(n-1)!}}\left.{\frac {d^{n}}{ds^{n}}}\left[s^{{n-1}}\log \xi (s)\right]\right|_{{s=1}}=\sum _{{\rho }}\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right],

where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of $|\Im (\rho )|$ .

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.

Hadamard product

A simple infinite product expansion is

\xi (s)={\frac {1}{2}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right),\!

where ρ ranges over the roots of ξ.

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 ρ and 1−ρ should be grouped together.

References

↑ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.

Further references

Weisstein, Eric W. "Xi-Function". MathWorld.
Keiper, J.B. (1992). "Power series expansions of Riemann's xi function". Mathematics of Computation. 58 (198): 765&ndash, 773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.

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[1] Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.