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 a Ξ {\displaystyle \Xi } (Greek uppercase letter "xi") by Edmund Landau. Landau's ξ {\displaystyle \xi } (lower-case "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, renamed as the 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 B n {\displaystyle B_{n}} denotes the n {\displaystyle 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
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