狄利克雷级数

$\sum_{n=1}^{\infty} \frac{a_n}{n^s},$

例子

$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$

$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$

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

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

$\zeta(s) \zeta(s-a)=\sum_{n=1}^{\infty} \frac{\sigma_{a}(n)}{n^s}$
$\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)} =\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}$

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

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

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

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

$\frac{\sigma_{1-s}(m)}{\zeta(s)}=\sum_{n=1}^\infty\frac{c_n(m)}{n^s}$

解析性质：收敛轴标

$f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$

导数

$F(s) =\sum_{n=1}^\infty \frac{f(n)}{n^s}$

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

乘积

$F(s)= \sum_{n=1}^{\infty} f(n)n^{-s}$

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

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

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