查看“︁赫尔维茨ζ函数”︁的源代码

[[File:Hurwitz zeta function.gif|thumb|300px|复空间赫尔维茨ζ函数]]
'''赫尔维茨ζ函数'''(Hurwitz zeta function)定义如下

:<math>\zeta(s,q) = \sum_{n=0}^\infty \frac{1}{(q+n)^{s}}.</math>

其中<math>q</math>、<math>s</math>都是复数，并且有<math>Re(q)>0</math>,<math>Re(s)>0</math>

对于给定的q,s,此函数可以扩展到 ''s''&ne;1的[[亚纯函数]].

[[黎曼ζ函数]]=<math>\zeta(s,1)</math>

==级数展开==
赫尔维茨ζ函数可以展开成级数：:<ref>{{Citation |first=Helmut |last=Hasse |title=Ein Summierungsverfahren für die Riemannsche ζ-Reihe |year=1930 |journal=[[Mathematische Zeitschrift]] |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 |jfm=56.0894.03 |url=https://eudml.org/doc/168238 |accessdate=2015-02-04 |archive-date=2017-08-05 |archive-url=https://web.archive.org/web/20170805014906/https://eudml.org/doc/168238 |dead-url=no }}</ref>


:<math>\zeta(s,q)=\frac{1}{s-1}
\sum_{n=0}^\infty \frac{1}{n+1}
\sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.</math>

此级数在S空间的[[紧空间]]子集中均匀收敛成为一个[[整函数]]。

==积分式==

赫尔维茨ζ函数可以表示为下列[[梅林变换]]

:<math>\zeta(s,q)=\frac{1}{\Gamma(s)} \int_0^\infty
\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt</math>

其中 <math>\Re s>1</math> 及<math>\Re q >0.</math>
==赫尔维茨公式==

:<math>\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]</math>
其中
:<math>\beta(x;s)=
2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}=
\frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix})
</math>

对于 <math>0\le x\le 1</math> and&nbsp;s&nbsp;>&nbsp;1成立，其中 <math>\text{Li}_s (z)</math>代表 [[多重对数]].

==泰勒展开==
赫尔维茨ζ函数的导数是平移：


:<math>\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).</math>

因此赫尔维茨ζ函数的[[泰勒级数]]可表示为:

:<math>\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).</math>

或

:<math>\zeta(s, q) = \frac{1}{q^s} + \sum_{n=0}^{\infty} (-q)^n {s + n - 1 \choose n} \zeta(s + n),</math>

其中 <math>|q| < 1</math>.<ref>{{cite arXiv |last=Vepsta卄s |first=Linas |authorlink= |eprint=math.CA/0702243 |title=An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions |class= |year=2007 |accessdate=29 August 2013 }}</ref>

==与Θ函數的关系==
 令<math>\vartheta (z,\tau)</math> 代表 雅可比 [[Θ函數]], 则

:<math>\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]</math>

对于 <math>\Re s > 0</math> and 复数''z'' 成立，但对于 ''z''=''n'' 整数，则有

:<math>\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).</math>

其中 ζ 代表[[黎曼ζ函数]].

==推广==
正整数m的赫尔维茨ζ函数与 [[多伽玛函数]]有下列关系:
:<math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z) \ .</math>
For negative integer −''n'' the values are related to the [[Bernoulli polynomials]]:<ref name=Ap264>Apostol (1976) p.264</ref>
:<math>\zeta(-n,x) = - \frac{B_{n+1}(x)}{n+1} \ . </math>

The [[巴恩斯ζ函数]]是赫尔维茨ζ函数的推广。

The [[勒奇超越函数]]也是赫尔维茨ζ函数的推广:
:<math>\Phi(z, s, q) = \sum_{k=0}^\infty
\frac { z^k} {(k+q)^s}</math>
即：
:<math>\zeta (s,q)=\Phi(1, s, q).\,</math>

赫尔维茨ζ函数与[[超几何函数]]的关系：

:<math>\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)</math>其中 <math>a_1=a_2=\ldots=a_s=a\text{ and }a\notin\N\text{ and }s\in\N^+.</math>

[[Meijer G函数]]

:<math>\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\N^+.</math>

==参考文献==
<references/>

==延伸阅读==
* {{cite book | last=Davenport | first=Harold | authorlink=Harold Davenport | title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
* {{cite journal
|first1=Jeff
|last1=Miller
|first2=Victor S.
|last2=Adamchik
|url=http://www-2.cs.cmu.edu/~adamchik/articles/hurwitz.htm
|title=Derivatives of the Hurwitz Zeta Function for Rational Arguments
|journal=Journal of Computational and Applied Mathematics
|volume=100
|year=1998
|pages=201–206
|doi=10.1016/S0377-0427(98)00193-9
|access-date=2015-02-04
|archive-date=2010-03-16
|archive-url=https://web.archive.org/web/20100316074311/http://www-2.cs.cmu.edu/~adamchik/articles/hurwitz.htm
|dead-url=no
}}
* {{cite web
|first1=Linas
|last1=Vepstas
|url=http://www.linas.org/math/gkw.pdf
|title=The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta
|access-date=2015-02-04
|archive-date=2021-03-10
|archive-url=https://web.archive.org/web/20210310053646/http://linas.org/math/gkw.pdf
|dead-url=no
}}
* {{cite journal
|first1=István
|last1=Mező
|first2=Ayhan
|last2=Dil
|doi=10.1016/j.jnt.2009.08.005
|title=Hyperharmonic series involving Hurwitz zeta function
|journal=Journal of Number Theory
|year=2010
|volume=130
|number=2
|pages=360–369
}}

[[Category:Ζ函數與L函數]]