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Riemann xi function

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Riemann xi function ξ ( s ) {\displaystyle \xi (s)} in the complex plane. The color of a point s {\displaystyle 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 {\displaystyle \xi } was renamed with an upper-case   Ξ   {\displaystyle ~\Xi ~} (Greek letter "Xi") by Edmund Landau. Landau's lower-case   ξ   {\displaystyle ~\xi ~} ("xi") is defined as

ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( s 2 ) ζ ( s ) {\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)}

for s ∈ C {\displaystyle s\in \mathbb {C} } . Here ζ ( s ) {\displaystyle \zeta (s)} denotes the Riemann zeta function and Γ ( s ) {\displaystyle \Gamma (s)} is the Gamma function.

The functional equation (or reflection formula) for Landau's   ξ   {\displaystyle ~\xi ~} is

ξ ( 1 − s ) = ξ ( s )   . {\displaystyle \xi (1-s)=\xi (s)~.}

Riemann's original function, rebaptised upper-case   Ξ   {\displaystyle ~\Xi ~} by Landau, satisfies

Ξ ( z ) = ξ ( 1 2 + z i ) {\displaystyle \Xi (z)=\xi \left({\tfrac {1}{2}}+zi\right)} ,

and obeys the functional equation

Ξ ( − z ) = Ξ ( z )   . {\displaystyle \Xi (-z)=\Xi (z)~.}

Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is

ξ ( 2 n ) = ( − 1 ) n + 1 n ! ( 2 n ) ! B 2 n 2 2 n − 1 π n ( 2 n − 1 ) {\displaystyle \xi (2n)=(-1)^{n+1}{\frac {n!}{(2n)!}}B_{2n}2^{2n-1}\pi ^{n}(2n-1)}

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

ξ ( 2 ) = π 6 {\displaystyle \xi (2)={\frac {\pi }{6}}}

Series representations

The ξ {\displaystyle \xi } function has the series expansion

d d z ln ⁡ ξ ( − z 1 − z ) = ∑ n = 0 ∞ λ n + 1 z n , {\displaystyle {\frac {d}{dz}}\ln \xi \left({\frac {-z}{1-z}}\right)=\sum _{n=0}^{\infty }\lambda _{n+1}z^{n},}

where

λ n = 1 ( n − 1 ) ! d n d s n [ s n − 1 log ⁡ ξ ( s ) ] | s = 1 = ∑ ρ [ 1 − ( 1 − 1 ρ ) n ] , {\displaystyle \lambda _{n}={\frac {1}{(n-1)!}}\left.{\frac {d^{n}}{ds^{n}}}\left\right|_{s=1}=\sum _{\rho }\left,}

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

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

Hadamard product

A simple infinite product expansion is

ξ ( s ) = 1 2 ∏ ρ ( 1 − s ρ ) , {\displaystyle \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

  1. ^ Landau, Edmund (1974) . 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.

This article incorporates material from Riemann Ξ function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Bernhard Riemann
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