# 狄利克雷级数

${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},}$

## 例子

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

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}$

${\displaystyle {\frac {1}{L(\chi ,s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)\chi (n)}{n^{s}}}}$

${\displaystyle {\frac {\zeta (s-1)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}}$

${\displaystyle \zeta (s)\zeta (s-a)=\sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}}$
${\displaystyle {\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}}=\sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}}$

${\displaystyle {\frac {\zeta ^{3}(s)}{\zeta (2s)}}=\sum _{n=1}^{\infty }{\frac {d(n^{2})}{n^{s}}}}$
${\displaystyle {\frac {\zeta ^{4}(s)}{\zeta (2s)}}=\sum _{n=1}^{\infty }{\frac {d(n)^{2}}{n^{s}}}}$

${\displaystyle \log \zeta (s)=\sum _{n=2}^{\infty }{\frac {\Lambda (n)}{\log(n)}}\,{\frac {1}{n^{s}}}}$

${\displaystyle {\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}.}$

${\displaystyle {\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}.}$

${\displaystyle {\frac {\sigma _{1-s}(m)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {c_{n}(m)}{n^{s}}}}$

## 解析性质：收敛轴标

${\displaystyle f(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}$

## 导数

${\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}}$

${\displaystyle {\frac {F^{\prime }(s)}{F(s)}}=-\sum _{n=1}^{\infty }{\frac {f(n)\Lambda (n)}{n^{s}}}}$

## 乘积

${\displaystyle F(s)=\sum _{n=1}^{\infty }f(n)n^{-s}}$

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

${\displaystyle T\sim \infty .}$时，${\displaystyle {\frac {1}{2T}}\int _{-T}^{T}\,dtF(a+it)G(b-it)\,dt=\sum _{n=1}^{\infty }f(n)g(n)n^{-a-b}}$

${\displaystyle T\sim \infty .}$时，${\displaystyle {\frac {1}{2T}}\int _{-T}^{T}dt|F(a+it)|^{2}dt=\sum _{n=1}^{\infty }[f(n)]^{2}n^{-2a}}$