# 赫尔维茨ζ函数

${\displaystyle \zeta (s,q)=\sum _{n=0}^{\infty }{\frac {1}{(q+n)^{s}}}.}$

## 级数展开

${\displaystyle \zeta (s,q)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(q+k)^{1-s}.}$

## 积分式

${\displaystyle \zeta (s,q)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-qt}}{1-e^{-t}}}dt}$

## 赫尔维茨公式

${\displaystyle \zeta (1-s,x)={\frac {1}{2s}}\left[e^{-i\pi s/2}\beta (x;s)+e^{i\pi s/2}\beta (1-x;s)\right]}$

${\displaystyle \beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi ix})}$

## 泰勒展开

${\displaystyle {\frac {\partial }{\partial q}}\zeta (s,q)=-s\zeta (s+1,q).}$

${\displaystyle \zeta (s,x+y)=\sum _{k=0}^{\infty }{\frac {y^{k}}{k!}}{\frac {\partial ^{k}}{\partial x^{k}}}\zeta (s,x)=\sum _{k=0}^{\infty }{s+k-1 \choose s-1}(-y)^{k}\zeta (s+k,x).}$

${\displaystyle \zeta (s,q)={\frac {1}{q^{s}}}+\sum _{n=0}^{\infty }(-q)^{n}{s+n-1 \choose n}\zeta (s+n),}$

## 与Θ函數的关系

令${\displaystyle \vartheta (z,\tau )}$  代表 雅可比 Θ函數, 则

${\displaystyle \int _{0}^{\infty }\left[\vartheta (z,it)-1\right]t^{s/2}{\frac {dt}{t}}=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\left[\zeta (1-s,z)+\zeta (1-s,1-z)\right]}$

${\displaystyle \int _{0}^{\infty }\left[\vartheta (n,it)-1\right]t^{s/2}{\frac {dt}{t}}=2\ \pi ^{-(1-s)/2}\ \Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)=2\ \pi ^{-s/2}\ \Gamma \left({\frac {s}{2}}\right)\zeta (s).}$

## 推广

${\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}m!\zeta (m+1,z)\ .}$

For negative integer −n the values are related to the Bernoulli polynomials:[3]

${\displaystyle \zeta (-n,x)=-{\frac {B_{n+1}(x)}{n+1}}\ .}$

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:

${\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}$

${\displaystyle \zeta (s,q)=\Phi (1,s,q).\,}$

${\displaystyle \zeta (s,a)=a^{-s}\cdot {}_{s+1}F_{s}(1,a_{1},a_{2},\ldots a_{s};a_{1}+1,a_{2}+1,\ldots a_{s}+1;1)}$ 其中 ${\displaystyle a_{1}=a_{2}=\ldots =a_{s}=a{\text{ and }}a\notin \mathbb {N} {\text{ and }}s\in \mathbb {N} ^{+}.}$
${\displaystyle \zeta (s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1\;\left|\;{\begin{matrix}0,1-a,\ldots ,1-a\\0,-a,\ldots ,-a\end{matrix}}\right)\right.\qquad \qquad s\in \mathbb {N} ^{+}.}$

## 参考文献

1. ^ Hasse, Helmut, Ein Summierungsverfahren für die Riemannsche ζ-Reihe, Mathematische Zeitschrift, 1930, 32 (1): 458–464 [2015-02-04], JFM 56.0894.03, doi:10.1007/BF01194645, （原始内容存档于2017-08-05）
2. ^ Vepsta卄s, Linas. An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions. 2007. . cite arXiv模板填写了不支持的参数 (帮助)
3. ^ Apostol (1976) p.264