# 狄利克雷η函数

${\displaystyle \eta (s)=\left(1-2^{1-s}\right)\zeta (s)}$

${\displaystyle \eta (s)=\sum _{n=1}^{\infty }{(-1)^{n-1} \over n^{s}}.}$

${\displaystyle \eta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s}}{\exp(x)+1}}{\frac {dx}{x}}}$

G·H·哈代给出一个函数方程的简单证明：

${\displaystyle \eta (-s)=2\pi ^{-s-1}s\sin \left({\pi s \over 2}\right)\Gamma (s)\eta (s+1).}$

## 数值算法

${\displaystyle \eta (s)=\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}{\frac {1}{(k+1)^{s}}}.}$

### Borwein方法

${\displaystyle d_{k}=n\sum _{i=0}^{k}{\frac {(n+i-1)!4^{i}}{(n-i)!(2i)!}}}$

${\displaystyle \eta (s)=-{\frac {1}{d_{n}}}\sum _{k=0}^{n-1}{\frac {(-1)^{k}(d_{k}-d_{n})}{(k+1)^{s}}}+\gamma _{n}(s),}$

${\displaystyle \Re (s)\geq {\frac {1}{2}}}$时，误差项 γn范围：

${\displaystyle |\gamma _{n}(s)|\leq {\frac {3}{(3+{\sqrt {8}})^{n}}}(1+2|\Im (s)|)\exp({\frac {\pi }{2}}|\Im (s)|).}$

## 特殊值

• η(0) = 12, 格兰迪级数（ 1 − 1 + 1 − 1 + · · ·）的阿贝尔和。
• η(−1) = 14, 1-2+3-4+…的阿贝尔和。
• 对于大于1的整数k ，如果Bk是第k伯努利数，那么
${\displaystyle \eta (1-k)={\frac {2^{k}-1}{k}}B_{k}.}$

${\displaystyle \!\ \eta (1)=\ln 2}$, 这是交错调和级数
${\displaystyle \eta (2)={\pi ^{2} \over 12}}$
${\displaystyle \eta (4)={{7\pi ^{4}} \over 720}}$
${\displaystyle \eta (6)={{31\pi ^{6}} \over 30240}}$
${\displaystyle \eta (8)={{127\pi ^{8}} \over 1209600}}$
${\displaystyle \eta (10)={{511\pi ^{10}} \over 6842880}}$
${\displaystyle \eta (12)={{1414477\pi ^{12}} \over {1307674368000}}}$

${\displaystyle \eta (2n)=(-1)^{n+1}{{B_{2n}\pi ^{2n}(2^{2n-1}-1)} \over {(2n)!}}.}$