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 original lower-case "xi"-function, $ξ$ was renamed with a $Ξ$ (Greek uppercase letter "xi") by Edmund Landau. Landau's $ξ$ (lower-case "xi") is defined as^[1]

ξ (s) = \frac{1}{2} s (s - 1) π^{- s / 2} Γ (\frac{s}{2}) ζ (s)

for $s \in ℂ$ . Here $ζ (s)$ denotes the Riemann zeta function and $Γ (s)$ is the gamma function.

The functional equation (or reflection formula) for Landau's $ξ$ is

ξ (1 - s) = ξ (s) .

Riemann's original function, renamed as the upper-case $Ξ$ by Landau,^[1] satisfies

Ξ (s) = ξ (\frac{1}{2} + s i),

and obeys the functional equation

Ξ (- s) = Ξ (s) .

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} \frac{n!}{(2 n)!} B_{2 n} 2^{2 n - 1} π^{n} (2 n - 1)

where $B_{n}$ denotes the ⁠ $n$ ⁠th Bernoulli number. For example:

ξ (2) = \frac{π}{6}

Series representations

The $ξ$ function has the series expansion

\frac{d}{d z} \ln ξ (\frac{- z}{1 - z}) = \sum_{n = 0}^{\infty} λ_{n + 1} z^{n},

where

λ_{n} = \frac{1}{(n - 1)!} {\frac{d^{n}}{d s^{n}} [s^{n - 1} \log ξ (s)] |}_{s = 1} = \sum_{ρ} [1 - {(1 - \frac{1}{ρ})}^{n}],

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

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

ξ (s) = \frac{1}{2} \prod_{ρ} (1 - \frac{s}{ρ}),

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 $\bar{ρ}$ should be grouped together.

References

1 2 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.

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.

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

[Landau1909-1] 1 2 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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