Riemann Xi function explained

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) = (1)/(2) s(s -1 ) \pi ^{-s/2} \Gamma\left( (s)/(2) \right) \zeta(s) for s \in 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 } + z i \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( 2 n) = ( -1 ) ^{n+1} (n!)/(( 2 n)!) B _{2n} 2 ^{2n-1} \pi ^{n} ( 2 n -1 )

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

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

Series representations

The

\xi function has the series expansion(d)/(dz) ln \xi \left( (- z)/(1- z) \right) = \sum ^{inf ty} _{n =0} λ _{n+1} z ^{n,}

where

λ _{n} = (1)/((n -1 )!) \left. (d ^{n})/(ds ^{n}) \left[s ^{n-1} log \xi(s) \right] \right| _{s=1} = \sum _{\rho} \left[ 1- \left( 1- (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) = (12)/(\prod) _{\rho} \left( 1 - (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

• • J.B. . Keiper . Mathematics of Computation . 1992 . 58 . 198 . 765 - 773 . Power series expansions of Riemann's xi function . 10.1090/S0025-5718-1992-1122072-5 . 1992MaCom..58..765K. free .

Notes and References

1. Book: Landau, Edmund . Edmund Landau . Handbuch der Lehre von der Verteilung der Primzahlen . Handbook of the Study of Distribution of the Prime Numbers . Teubner . Leipzig --> . 1909 . Third . Chelsea . New York . 1974 . §70-71 and page 894.