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, $\xi$ was renamed with an upper-case $~\Xi ~$ (Greek letter "Xi") by Edmund Landau. Landau's lower-case $~\xi ~$ ("xi") is defined as^[1]

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

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

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

Riemann's original function, rebaptised upper-case $~\Xi ~$ by Landau,^[1] satisfies

\Xi (z)=\xi \left({\tfrac {1}{2}}+zi\right)

,

and obeys the functional equation

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

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

^ ^a ^b Landau, Edmund (1974) [1909]. Handbuch der Lehre von der Verteilung der Primzahlen [Handbook of the Study of Distribution of the Prime Numbers] (Third ed.). New York: Chelsea. §70-71 and page 894.

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): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.

[Landau1909-1] Landau, Edmund (1974) [1909]. Handbuch der Lehre von der Verteilung der Primzahlen [Handbook of the Study of Distribution of the Prime Numbers] (Third ed.). New York: Chelsea. §70-71 and page 894.

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