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Riemann Xi function

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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.

Contents

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:

ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( 1 2 s ) ζ ( s )

for s ∈ 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

ξ ( 1 − s ) = ξ ( s ) .

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

Ξ ( z ) = ξ ( 1 2 + z i )

and obeys the functional equation

Ξ ( − z ) = Ξ ( z ) .

As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ. Both would be entire functions if you filled every removable singularity in.

Values

The general form for even integers is

ξ ( 2 n ) = ( − 1 ) n + 1 n ! ( 2 n ) ! B 2 n 2 2 n − 1 π n ( 2 n − 1 )

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

ξ ( 2 ) = π 6

Series representations

The ξ function has the series expansion

d d z ln ⁡ ξ ( − z 1 − z ) = ∑ n = 0 ∞ λ 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 ] ,

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 λn > 0 for all positive n.

Hadamard product

A simple infinite product expansion is

Ξ ( s ) = Ξ ( 0 ) ∏ ρ ( 1 − 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 1−ρ should be grouped together.

References

Riemann Xi function Wikipedia
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