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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[1]
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,[1] 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.
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 denotes the n -th Bernoulli number . For example:
ξ
(
2
)
=
π
6
{\displaystyle \xi (2 )={\frac {\pi }{6}}}
Series representations [ edit ]
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[s^{n-1}\log \xi (s )\right]\right|_{s=1}=\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right],}
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 [ edit ]
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 [ edit ]
^ a b 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.
This article incorporates material from Riemann Ξ function on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Riemann_Xi_function&oldid=1089425365 "
C a t e g o r i e s :
● Z e t a a n d L - f u n c t i o n s
● B e r n h a r d R i e m a n n
H i d d e n c a t e g o r y :
● W i k i p e d i a a r t i c l e s i n c o r p o r a t i n g t e x t f r o m P l a n e t M a t h
● T h i s p a g e w a s l a s t e d i t e d o n 2 3 M a y 2 0 2 2 , a t 1 8 : 3 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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