Special function in mathematics
In mathematics , the Hurwitz zeta function is one of the many zeta functions . It is formally defined for complex variables s with Re(s ) > 1 and a ≠ 0, −1, −2, … by
ζ ぜーた
(
s
,
a
)
=
∑
n
=
0
∞
1
(
n
+
a
)
s
.
{\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.}
This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1 . The Riemann zeta function is ζ ぜーた (s ,1) . The Hurwitz zeta function is named after Adolf Hurwitz , who introduced it in 1882.[1]
Hurwitz zeta function corresponding to a = 1/3 . It is generated as a Matplotlib plot using a version of the Domain coloring method.[2]
Hurwitz zeta function corresponding to a = 24/25 .
Hurwitz zeta function as a function of a with s = 3 + 4i .
Integral representation [ edit ]
The Hurwitz zeta function has an integral representation
ζ ぜーた
(
s
,
a
)
=
1
Γ がんま
(
s
)
∫
0
∞
x
s
−
1
e
−
a
x
1
−
e
−
x
d
x
{\displaystyle \zeta (s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-ax}}{1-e^{-x}}}dx}
for
Re
(
s
)
>
1
{\displaystyle \operatorname {Re} (s)>1}
and
Re
(
a
)
>
0.
{\displaystyle \operatorname {Re} (a)>0.}
(This integral can be viewed as a Mellin transform .) The formula can be obtained, roughly, by writing
ζ ぜーた
(
s
,
a
)
Γ がんま
(
s
)
=
∑
n
=
0
∞
1
(
n
+
a
)
s
∫
0
∞
x
s
e
−
x
d
x
x
=
∑
n
=
0
∞
∫
0
∞
y
s
e
−
(
n
+
a
)
y
d
y
y
{\displaystyle \zeta (s,a)\Gamma (s)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}=\sum _{n=0}^{\infty }\int _{0}^{\infty }y^{s}e^{-(n+a)y}{\frac {dy}{y}}}
and then interchanging the sum and integral.[3]
The integral representation above can be converted to a contour integral representation
ζ ぜーた
(
s
,
a
)
=
−
Γ がんま
(
1
−
s
)
1
2
π ぱい
i
∫
C
(
−
z
)
s
−
1
e
−
a
z
1
−
e
−
z
d
z
{\displaystyle \zeta (s,a)=-\Gamma (1-s){\frac {1}{2\pi i}}\int _{C}{\frac {(-z)^{s-1}e^{-az}}{1-e^{-z}}}dz}
where
C
{\displaystyle C}
is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation
(
−
z
)
s
−
1
{\displaystyle (-z)^{s-1}}
. Unlike the previous integral, this integral is valid for all s , and indeed is an entire function of s .[4]
The contour integral representation provides an analytic continuation of
ζ ぜーた
(
s
,
a
)
{\displaystyle \zeta (s,a)}
to all
s
≠
1
{\displaystyle s\neq 1}
. At
s
=
1
{\displaystyle s=1}
, it has a simple pole with residue
1
{\displaystyle 1}
.[5]
Hurwitz's formula [ edit ]
The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function :[6]
ζ ぜーた
(
1
−
s
,
a
)
=
Γ がんま
(
s
)
(
2
π ぱい
)
s
(
e
−
π ぱい
i
s
/
2
∑
n
=
1
∞
e
2
π ぱい
i
n
a
n
s
+
e
π ぱい
i
s
/
2
∑
n
=
1
∞
e
−
2
π ぱい
i
n
a
n
s
)
,
{\displaystyle \zeta (1-s,a)={\frac {\Gamma (s)}{(2\pi )^{s}}}\left(e^{-\pi is/2}\sum _{n=1}^{\infty }{\frac {e^{2\pi ina}}{n^{s}}}+e^{\pi is/2}\sum _{n=1}^{\infty }{\frac {e^{-2\pi ina}}{n^{s}}}\right),}
valid for Re(s ) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:[7]
ζ ぜーた
(
1
−
s
)
=
2
Γ がんま
(
s
)
(
2
π ぱい
)
s
cos
(
π ぱい
s
2
)
ζ ぜーた
(
s
)
{\displaystyle \zeta (1-s)={\frac {2\Gamma (s)}{(2\pi )^{s}}}\cos \left({\frac {\pi s}{2}}\right)\zeta (s)}
Hurwitz's formula can also be expressed as[8]
ζ ぜーた
(
s
,
a
)
=
2
Γ がんま
(
1
−
s
)
(
2
π ぱい
)
1
−
s
(
sin
(
π ぱい
s
2
)
∑
n
=
1
∞
cos
(
2
π ぱい
n
a
)
n
1
−
s
+
cos
(
π ぱい
s
2
)
∑
n
=
1
∞
sin
(
2
π ぱい
n
a
)
n
1
−
s
)
{\displaystyle \zeta (s,a)={\frac {2\Gamma (1-s)}{(2\pi )^{1-s}}}\left(\sin \left({\frac {\pi s}{2}}\right)\sum _{n=1}^{\infty }{\frac {\cos(2\pi na)}{n^{1-s}}}+\cos \left({\frac {\pi s}{2}}\right)\sum _{n=1}^{\infty }{\frac {\sin(2\pi na)}{n^{1-s}}}\right)}
(for Re(s ) < 0 and 0 < a ≤ 1).
Hurwitz's formula has a variety of different proofs.[9] One proof uses the contour integration representation along with the residue theorem .[6] [8] A second proof uses a theta function identity, or equivalently Poisson summation .[10] These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper . Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral
ζ ぜーた
(
s
,
a
)
=
s
∫
−
a
∞
⌊
x
⌋
−
x
+
1
2
(
x
+
a
)
s
+
1
d
x
{\displaystyle \zeta (s,a)=s\int _{-a}^{\infty }{\frac {\lfloor x\rfloor -x+{\frac {1}{2}}}{(x+a)^{s+1}}}dx}
(−1 < Re(s ) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series .[11]
Functional equation for rational a [ edit ]
When a is a rational number, Hurwitz's formula leads to the following functional equation : For integers
1
≤
m
≤
n
{\displaystyle 1\leq m\leq n}
,
ζ ぜーた
(
1
−
s
,
m
n
)
=
2
Γ がんま
(
s
)
(
2
π ぱい
n
)
s
∑
k
=
1
n
[
cos
(
π ぱい
s
2
−
2
π ぱい
k
m
n
)
ζ ぜーた
(
s
,
k
n
)
]
{\displaystyle \zeta \left(1-s,{\frac {m}{n}}\right)={\frac {2\Gamma (s)}{(2\pi n)^{s}}}\sum _{k=1}^{n}\left[\cos \left({\frac {\pi s}{2}}-{\frac {2\pi km}{n}}\right)\;\zeta \left(s,{\frac {k}{n}}\right)\right]}
holds for all values of s .[12]
This functional equation can be written as another equivalent form:
ζ ぜーた
(
1
−
s
,
m
n
)
=
Γ がんま
(
s
)
(
2
π ぱい
n
)
s
∑
k
=
1
n
[
e
π ぱい
i
s
2
e
−
2
π ぱい
i
k
m
n
ζ ぜーた
(
s
,
k
n
)
+
e
−
π ぱい
i
s
2
e
2
π ぱい
i
k
m
n
ζ ぜーた
(
s
,
k
n
)
]
{\displaystyle \zeta \left(1-s,{\frac {m}{n}}\right)={\frac {\Gamma (s)}{(2\pi n)^{s}}}\sum _{k=1}^{n}\left[e^{\frac {\pi is}{2}}e^{-{\frac {2\pi ikm}{n}}}\zeta \left(s,{\frac {k}{n}}\right)+e^{-{\frac {\pi is}{2}}}e^{\frac {2\pi ikm}{n}}\zeta \left(s,{\frac {k}{n}}\right)\right]}
.
Some finite sums [ edit ]
Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form
∑
r
=
1
m
−
1
ζ ぜーた
(
s
,
r
m
)
cos
2
π ぱい
r
k
m
=
m
Γ がんま
(
1
−
s
)
(
2
π ぱい
m
)
1
−
s
sin
π ぱい
s
2
⋅
{
ζ ぜーた
(
1
−
s
,
k
m
)
+
ζ ぜーた
(
1
−
s
,
1
−
k
m
)
}
−
ζ ぜーた
(
s
)
{\displaystyle \sum _{r=1}^{m-1}\zeta \left(s,{\frac {r}{m}}\right)\cos {\dfrac {2\pi rk}{m}}={\frac {m\Gamma (1-s)}{(2\pi m)^{1-s}}}\sin {\frac {\pi s}{2}}\cdot \left\{\zeta \left(1-s,{\frac {k}{m}}\right)+\zeta \left(1-s,1-{\frac {k}{m}}\right)\right\}-\zeta (s)}
∑
r
=
1
m
−
1
ζ ぜーた
(
s
,
r
m
)
sin
2
π ぱい
r
k
m
=
m
Γ がんま
(
1
−
s
)
(
2
π ぱい
m
)
1
−
s
cos
π ぱい
s
2
⋅
{
ζ ぜーた
(
1
−
s
,
k
m
)
−
ζ ぜーた
(
1
−
s
,
1
−
k
m
)
}
{\displaystyle \sum _{r=1}^{m-1}\zeta \left(s,{\frac {r}{m}}\right)\sin {\dfrac {2\pi rk}{m}}={\frac {m\Gamma (1-s)}{(2\pi m)^{1-s}}}\cos {\frac {\pi s}{2}}\cdot \left\{\zeta \left(1-s,{\frac {k}{m}}\right)-\zeta \left(1-s,1-{\frac {k}{m}}\right)\right\}}
∑
r
=
1
m
−
1
ζ ぜーた
2
(
s
,
r
m
)
=
(
m
2
s
−
1
−
1
)
ζ ぜーた
2
(
s
)
+
2
m
Γ がんま
2
(
1
−
s
)
(
2
π ぱい
m
)
2
−
2
s
∑
l
=
1
m
−
1
{
ζ ぜーた
(
1
−
s
,
l
m
)
−
cos
π ぱい
s
⋅
ζ ぜーた
(
1
−
s
,
1
−
l
m
)
}
ζ ぜーた
(
1
−
s
,
l
m
)
{\displaystyle \sum _{r=1}^{m-1}\zeta ^{2}\left(s,{\frac {r}{m}}\right)={\big (}m^{2s-1}-1{\big )}\zeta ^{2}(s)+{\frac {2m\Gamma ^{2}(1-s)}{(2\pi m)^{2-2s}}}\sum _{l=1}^{m-1}\left\{\zeta \left(1-s,{\frac {l}{m}}\right)-\cos \pi s\cdot \zeta \left(1-s,1-{\frac {l}{m}}\right)\right\}\zeta \left(1-s,{\frac {l}{m}}\right)}
where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.[13]
Series representation [ edit ]
A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:[14]
ζ ぜーた
(
s
,
a
)
=
1
s
−
1
∑
n
=
0
∞
1
n
+
1
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
a
+
k
)
1
−
s
.
{\displaystyle \zeta (s,a)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(a+k)^{1-s}.}
This series converges uniformly on compact subsets of the s -plane to an entire function . The inner sum may be understood to be the n th forward difference of
a
1
−
s
{\displaystyle a^{1-s}}
; that is,
Δ でるた
n
a
1
−
s
=
∑
k
=
0
n
(
−
1
)
n
−
k
(
n
k
)
(
a
+
k
)
1
−
s
{\displaystyle \Delta ^{n}a^{1-s}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(a+k)^{1-s}}
where Δ でるた is the forward difference operator . Thus, one may write:
ζ ぜーた
(
s
,
a
)
=
1
s
−
1
∑
n
=
0
∞
(
−
1
)
n
n
+
1
Δ でるた
n
a
1
−
s
=
1
s
−
1
log
(
1
+
Δ でるた
)
Δ でるた
a
1
−
s
{\displaystyle {\begin{aligned}\zeta (s,a)&={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n+1}}\Delta ^{n}a^{1-s}\\&={\frac {1}{s-1}}{\log(1+\Delta ) \over \Delta }a^{1-s}\end{aligned}}}
Taylor series [ edit ]
The partial derivative of the zeta in the second argument is a shift :
∂
∂
a
ζ ぜーた
(
s
,
a
)
=
−
s
ζ ぜーた
(
s
+
1
,
a
)
.
{\displaystyle {\frac {\partial }{\partial a}}\zeta (s,a)=-s\zeta (s+1,a).}
Thus, the Taylor series can be written as:
ζ ぜーた
(
s
,
x
+
y
)
=
∑
k
=
0
∞
y
k
k
!
∂
k
∂
x
k
ζ ぜーた
(
s
,
x
)
=
∑
k
=
0
∞
(
s
+
k
−
1
s
−
1
)
(
−
y
)
k
ζ ぜーた
(
s
+
k
,
x
)
.
{\displaystyle \zeta (s,x+y)=\sum _{k=0}^{\infty }{\frac {y^{k}}{k!}}{\frac {\partial ^{k}}{\partial x^{k}}}\zeta (s,x)=\sum _{k=0}^{\infty }{s+k-1 \choose s-1}(-y)^{k}\zeta (s+k,x).}
Alternatively,
ζ ぜーた
(
s
,
q
)
=
1
q
s
+
∑
n
=
0
∞
(
−
q
)
n
(
s
+
n
−
1
n
)
ζ ぜーた
(
s
+
n
)
,
{\displaystyle \zeta (s,q)={\frac {1}{q^{s}}}+\sum _{n=0}^{\infty }(-q)^{n}{s+n-1 \choose n}\zeta (s+n),}
with
|
q
|
<
1
{\displaystyle |q|<1}
.[15]
Closely related is the Stark–Keiper formula:
ζ ぜーた
(
s
,
N
)
=
∑
k
=
0
∞
[
N
+
s
−
1
k
+
1
]
(
s
+
k
−
1
s
−
1
)
(
−
1
)
k
ζ ぜーた
(
s
+
k
,
N
)
{\displaystyle \zeta (s,N)=\sum _{k=0}^{\infty }\left[N+{\frac {s-1}{k+1}}\right]{s+k-1 \choose s-1}(-1)^{k}\zeta (s+k,N)}
which holds for integer N and arbitrary s . See also Faulhaber's formula for a similar relation on finite sums of powers of integers.
Laurent series [ edit ]
The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series
ζ ぜーた
(
s
,
a
)
=
1
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
n
!
γ がんま
n
(
a
)
(
s
−
1
)
n
.
{\displaystyle \zeta (s,a)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}(a)(s-1)^{n}.}
In particular, the constant term is given by
lim
s
→
1
[
ζ ぜーた
(
s
,
a
)
−
1
s
−
1
]
=
−
Γ がんま
′
(
a
)
Γ がんま
(
a
)
=
−
ψ ぷさい
(
a
)
{\displaystyle \lim _{s\to 1}\left[\zeta (s,a)-{\frac {1}{s-1}}\right]={\frac {-\Gamma '(a)}{\Gamma (a)}}=-\psi (a)}
where
Γ がんま
{\displaystyle \Gamma }
is the gamma function and
ψ ぷさい
=
Γ がんま
′
/
Γ がんま
{\displaystyle \psi =\Gamma '/\Gamma }
is the digamma function . As a special case,
γ がんま
0
(
1
)
=
−
ψ ぷさい
(
1
)
=
γ がんま
0
=
γ がんま
{\displaystyle \gamma _{0}(1)=-\psi (1)=\gamma _{0}=\gamma }
.
Discrete Fourier transform [ edit ]
The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function .[16]
Particular values [ edit ]
Negative integers [ edit ]
The values of ζ ぜーた (s , a ) at s = 0, −1, −2, ... are related to the Bernoulli polynomials :[17]
ζ ぜーた
(
−
n
,
a
)
=
−
B
n
+
1
(
a
)
n
+
1
.
{\displaystyle \zeta (-n,a)=-{\frac {B_{n+1}(a)}{n+1}}.}
For example, the
n
=
0
{\displaystyle n=0}
case gives[18]
ζ ぜーた
(
0
,
a
)
=
1
2
−
a
.
{\displaystyle \zeta (0,a)={\frac {1}{2}}-a.}
s -derivative[ edit ]
The partial derivative with respect to s at s = 0 is related to the gamma function:
∂
∂
s
ζ ぜーた
(
s
,
a
)
|
s
=
0
=
log
Γ がんま
(
a
)
−
1
2
log
(
2
π ぱい
)
{\displaystyle \left.{\frac {\partial }{\partial s}}\zeta (s,a)\right|_{s=0}=\log \Gamma (a)-{\frac {1}{2}}\log(2\pi )}
In particular,
ζ ぜーた
′
(
0
)
=
−
1
2
log
(
2
π ぱい
)
.
{\textstyle \zeta '(0)=-{\frac {1}{2}}\log(2\pi ).}
The formula is due to Lerch .[19] [20]
Relation to Jacobi theta function [ edit ]
If
ϑ
(
z
,
τ たう
)
{\displaystyle \vartheta (z,\tau )}
is the Jacobi theta function , then
∫
0
∞
[
ϑ
(
z
,
i
t
)
−
1
]
t
s
/
2
d
t
t
=
π ぱい
−
(
1
−
s
)
/
2
Γ がんま
(
1
−
s
2
)
[
ζ ぜーた
(
1
−
s
,
z
)
+
ζ ぜーた
(
1
−
s
,
1
−
z
)
]
{\displaystyle \int _{0}^{\infty }\left[\vartheta (z,it)-1\right]t^{s/2}{\frac {dt}{t}}=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\left[\zeta (1-s,z)+\zeta (1-s,1-z)\right]}
holds for
ℜ
s
>
0
{\displaystyle \Re s>0}
and z complex, but not an integer. For z =n an integer, this simplifies to
∫
0
∞
[
ϑ
(
n
,
i
t
)
−
1
]
t
s
/
2
d
t
t
=
2
π ぱい
−
(
1
−
s
)
/
2
Γ がんま
(
1
−
s
2
)
ζ ぜーた
(
1
−
s
)
=
2
π ぱい
−
s
/
2
Γ がんま
(
s
2
)
ζ ぜーた
(
s
)
.
{\displaystyle \int _{0}^{\infty }\left[\vartheta (n,it)-1\right]t^{s/2}{\frac {dt}{t}}=2\ \pi ^{-(1-s)/2}\ \Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)=2\ \pi ^{-s/2}\ \Gamma \left({\frac {s}{2}}\right)\zeta (s).}
where ζ ぜーた here is the Riemann zeta function . Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function , or Dirac comb in z as
t
→
0
{\displaystyle t\rightarrow 0}
.
Relation to Dirichlet L -functions [ edit ]
At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ ぜーた (s ) when a = 1, when a = 1/2 it is equal to (2s −1)ζ ぜーた (s ),[21] and if a = n /k with k > 2, (n ,k ) > 1 and 0 < n < k , then[22]
ζ ぜーた
(
s
,
n
/
k
)
=
k
s
φ ふぁい
(
k
)
∑
χ かい
χ かい
¯
(
n
)
L
(
s
,
χ かい
)
,
{\displaystyle \zeta (s,n/k)={\frac {k^{s}}{\varphi (k)}}\sum _{\chi }{\overline {\chi }}(n)L(s,\chi ),}
the sum running over all Dirichlet characters mod k . In the opposite direction we have the linear combination[21]
L
(
s
,
χ かい
)
=
1
k
s
∑
n
=
1
k
χ かい
(
n
)
ζ ぜーた
(
s
,
n
k
)
.
{\displaystyle L(s,\chi )={\frac {1}{k^{s}}}\sum _{n=1}^{k}\chi (n)\;\zeta \left(s,{\frac {n}{k}}\right).}
There is also the multiplication theorem
k
s
ζ ぜーた
(
s
)
=
∑
n
=
1
k
ζ ぜーた
(
s
,
n
k
)
,
{\displaystyle k^{s}\zeta (s)=\sum _{n=1}^{k}\zeta \left(s,{\frac {n}{k}}\right),}
of which a useful generalization is the distribution relation [23]
∑
p
=
0
q
−
1
ζ ぜーた
(
s
,
a
+
p
/
q
)
=
q
s
ζ ぜーた
(
s
,
q
a
)
.
{\displaystyle \sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa).}
(This last form is valid whenever q a natural number and 1 − qa is not.)
If a =1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if a =1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra ), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<a <1 and a ≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(s )<1+ε いぷしろん for any positive real number ε いぷしろん . This was proved by Davenport and Heilbronn for rational or transcendental irrational a ,[24] and by Cassels for algebraic irrational a .[21] [25]
Rational values [ edit ]
The Hurwitz zeta function occurs in a number of striking identities at rational values.[26] In particular, values in terms of the Euler polynomials
E
n
(
x
)
{\displaystyle E_{n}(x)}
:
E
2
n
−
1
(
p
q
)
=
(
−
1
)
n
4
(
2
n
−
1
)
!
(
2
π ぱい
q
)
2
n
∑
k
=
1
q
ζ ぜーた
(
2
n
,
2
k
−
1
2
q
)
cos
(
2
k
−
1
)
π ぱい
p
q
{\displaystyle E_{2n-1}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n-1)!}{(2\pi q)^{2n}}}\sum _{k=1}^{q}\zeta \left(2n,{\frac {2k-1}{2q}}\right)\cos {\frac {(2k-1)\pi p}{q}}}
and
E
2
n
(
p
q
)
=
(
−
1
)
n
4
(
2
n
)
!
(
2
π ぱい
q
)
2
n
+
1
∑
k
=
1
q
ζ ぜーた
(
2
n
+
1
,
2
k
−
1
2
q
)
sin
(
2
k
−
1
)
π ぱい
p
q
{\displaystyle E_{2n}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n)!}{(2\pi q)^{2n+1}}}\sum _{k=1}^{q}\zeta \left(2n+1,{\frac {2k-1}{2q}}\right)\sin {\frac {(2k-1)\pi p}{q}}}
One also has
ζ ぜーた
(
s
,
2
p
−
1
2
q
)
=
2
(
2
q
)
s
−
1
∑
k
=
1
q
[
C
s
(
k
q
)
cos
(
(
2
p
−
1
)
π ぱい
k
q
)
+
S
s
(
k
q
)
sin
(
(
2
p
−
1
)
π ぱい
k
q
)
]
{\displaystyle \zeta \left(s,{\frac {2p-1}{2q}}\right)=2(2q)^{s-1}\sum _{k=1}^{q}\left[C_{s}\left({\frac {k}{q}}\right)\cos \left({\frac {(2p-1)\pi k}{q}}\right)+S_{s}\left({\frac {k}{q}}\right)\sin \left({\frac {(2p-1)\pi k}{q}}\right)\right]}
which holds for
1
≤
p
≤
q
{\displaystyle 1\leq p\leq q}
. Here, the
C
ν にゅー
(
x
)
{\displaystyle C_{\nu }(x)}
and
S
ν にゅー
(
x
)
{\displaystyle S_{\nu }(x)}
are defined by means of the Legendre chi function
χ かい
ν にゅー
{\displaystyle \chi _{\nu }}
as
C
ν にゅー
(
x
)
=
Re
χ かい
ν にゅー
(
e
i
x
)
{\displaystyle C_{\nu }(x)=\operatorname {Re} \,\chi _{\nu }(e^{ix})}
and
S
ν にゅー
(
x
)
=
Im
χ かい
ν にゅー
(
e
i
x
)
.
{\displaystyle S_{\nu }(x)=\operatorname {Im} \,\chi _{\nu }(e^{ix}).}
For integer values of ν にゅー , these may be expressed in terms of the Euler polynomials. These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.
Applications [ edit ]
Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory , where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems . In applied statistics , it occurs in Zipf's law and the Zipf–Mandelbrot law . In particle physics , it occurs in a formula by Julian Schwinger ,[27] giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.
Special cases and generalizations [ edit ]
The Hurwitz zeta function with a positive integer m is related to the polygamma function :
ψ ぷさい
(
m
)
(
z
)
=
(
−
1
)
m
+
1
m
!
ζ ぜーた
(
m
+
1
,
z
)
.
{\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}m!\zeta (m+1,z)\ .}
The Barnes zeta function generalizes the Hurwitz zeta function.
The Lerch transcendent generalizes the Hurwitz zeta:
Φ ふぁい
(
z
,
s
,
q
)
=
∑
k
=
0
∞
z
k
(
k
+
q
)
s
{\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}
and thus
ζ ぜーた
(
s
,
a
)
=
Φ ふぁい
(
1
,
s
,
a
)
.
{\displaystyle \zeta (s,a)=\Phi (1,s,a).\,}
Hypergeometric function
ζ ぜーた
(
s
,
a
)
=
a
−
s
⋅
s
+
1
F
s
(
1
,
a
1
,
a
2
,
…
a
s
;
a
1
+
1
,
a
2
+
1
,
…
a
s
+
1
;
1
)
{\displaystyle \zeta (s,a)=a^{-s}\cdot {}_{s+1}F_{s}(1,a_{1},a_{2},\ldots a_{s};a_{1}+1,a_{2}+1,\ldots a_{s}+1;1)}
where
a
1
=
a
2
=
…
=
a
s
=
a
and
a
∉
N
and
s
∈
N
+
.
{\displaystyle a_{1}=a_{2}=\ldots =a_{s}=a{\text{ and }}a\notin \mathbb {N} {\text{ and }}s\in \mathbb {N} ^{+}.}
Meijer G-function
ζ ぜーた
(
s
,
a
)
=
G
s
+
1
,
s
+
1
1
,
s
+
1
(
−
1
|
0
,
1
−
a
,
…
,
1
−
a
0
,
−
a
,
…
,
−
a
)
s
∈
N
+
.
{\displaystyle \zeta (s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1\;\left|\;{\begin{matrix}0,1-a,\ldots ,1-a\\0,-a,\ldots ,-a\end{matrix}}\right)\right.\qquad \qquad s\in \mathbb {N} ^{+}.}
^ Hurwitz, Adolf (1882). "Einige Eigenschaften der Dirichlet'schen Functionen
F
(
s
)
=
∑
(
D
n
)
⋅
1
n
{\textstyle F(s)=\sum \left({\frac {D}{n}}\right)\cdot {\frac {1}{n}}}
, die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten" . Zeitschrift für Mathematik und Physik (in German). 27 : 86–101.
^ "Jupyter Notebook Viewer" .
^ Apostol 1976 , p. 251, Theorem 12.2
^ Whittaker & Watson 1927 , p. 266, Section 13.13
^ Apostol 1976 , p. 255, Theorem 12.4
^ a b Apostol 1976 , p. 257, Theorem 12.6
^ Apostol 1976 , p. 259, Theorem 12.7
^ a b Whittaker & Watson 1927 , pp. 268–269, Section 13.15
^ See the references in Section 4 of: Kanemitsu, S.; Tanigawa, Y.; Tsukada, H.; Yoshimoto, M. (2007). "Contributions to the theory of the Hurwitz zeta-function" . Hardy-Ramanujan Journal . 30 : 31–55. doi :10.46298/hrj.2007.159 . Zbl 1157.11036 .
^ Fine, N. J. (June 1951). "Note on the Hurwitz Zeta-Function" . Proceedings of the American Mathematical Society . 2 (3): 361–364. doi :10.2307/2031757 . JSTOR 2031757 . Zbl 0043.07802 .
^ Berndt, Bruce C. (Winter 1972). "On the Hurwitz zeta-function" . Rocky Mountain Journal of Mathematics . 2 (1): 151–158. doi :10.1216/RMJ-1972-2-1-151 . Zbl 0229.10023 .
^ Apostol 1976 , p. 261, Theorem 12.8
^ Blagouchine, I.V. (2014). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory . 148 . Elsevier: 537–592. arXiv :1401.3724 . doi :10.1016/j.jnt.2014.08.009 .
^ Hasse, Helmut (1930), "Ein Summierungsverfahren für die Riemannsche ζ ぜーた -Reihe" , Mathematische Zeitschrift , 32 (1): 458–464, doi :10.1007/BF01194645 , JFM 56.0894.03 , S2CID 120392534
^ Vepstas, Linas (2007). "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions". Numerical Algorithms . 47 (3): 211–252. arXiv :math/0702243 . Bibcode :2008NuAlg..47..211V . doi :10.1007/s11075-007-9153-8 . S2CID 15131811 .
^ Jacek Klinowski, Djurdje Cvijović (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments" . Mathematics of Computation . 68 (228): 1623–1631. Bibcode :1999MaCom..68.1623C . doi :10.1090/S0025-5718-99-01091-1 .
^ Apostol 1976 , p. 264, Theorem 12.13
^ Apostol 1976 , p. 268
^ Berndt, Bruce C. (1985). "The Gamma Function and the Hurwitz Zeta-Function". The American Mathematical Monthly . 92 (2): 126–130. doi :10.2307/2322640 . JSTOR 2322640 .
^ Whittaker & Watson 1927 , p. 271, Section 13.21
^ a b c Davenport (1967) p.73
^ Lowry, David (8 February 2013). "Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa" . mixedmath . Retrieved 8 February 2013 .
^ Kubert, Daniel S. ; Lang, Serge (1981). Modular Units . Grundlehren der Mathematischen Wissenschaften. Vol. 244. Springer-Verlag . p. 13. ISBN 0-387-90517-0 . Zbl 0492.12002 .
^ Davenport, H. & Heilbronn, H. (1936), "On the zeros of certain Dirichlet series", Journal of the London Mathematical Society , 11 (3): 181–185, doi :10.1112/jlms/s1-11.3.181 , Zbl 0014.21601
^ Cassels, J. W. S. (1961), "Footnote to a note of Davenport and Heilbronn", Journal of the London Mathematical Society , 36 (1): 177–184, doi :10.1112/jlms/s1-36.1.177 , Zbl 0097.03403
^ Given by Cvijović, Djurdje & Klinowski, Jacek (1999), "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation , 68 (228): 1623–1630, Bibcode :1999MaCom..68.1623C , doi :10.1090/S0025-5718-99-01091-1
^ Schwinger, J. (1951), "On gauge invariance and vacuum polarization", Physical Review , 82 (5): 664–679, Bibcode :1951PhRv...82..664S , doi :10.1103/PhysRev.82.664
References [ edit ]
Apostol, T. M. (2010), "Hurwitz zeta function" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .
See chapter 12 of Apostol, Tom M. (1976), Introduction to analytic number theory , Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3 , MR 0434929 , Zbl 0335.10001
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions , (1964) Dover Publications, New York. ISBN 0-486-61272-4 . (See Paragraph 6.4.10 for relationship to polygamma function.)
Davenport, Harold (1967). Multiplicative number theory . Lectures in advanced mathematics. Vol. 1. Chicago: Markham. Zbl 0159.06303 .
Miller, Jeff; Adamchik, Victor S. (1998). "Derivatives of the Hurwitz Zeta Function for Rational Arguments" . Journal of Computational and Applied Mathematics . 100 (2): 201–206. doi :10.1016/S0377-0427(98)00193-9 .
Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory . 130 (2): 360–369. doi :10.1016/j.jnt.2009.08.005 . hdl :2437/90539 .
Whittaker, E. T. ; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press .
External links [ edit ]