# Formelsammlung Mathematik: Unendliche Reihen: Dirichletreihen

Zurück zu Unendliche Reihen

##### 1
${\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}\qquad {\text{Re}}(s)>2}$

##### 2
${\displaystyle \sum _{n=1}^{\infty }{\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\,\zeta (s-1)}{\zeta (2s)}}\qquad {\text{Re}}(s)>2}$

##### 3
${\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}\qquad {\text{Re}}(s)>1}$

##### 4
${\displaystyle \sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}=-{\frac {\zeta '(s)}{\zeta (s)}}\qquad {\text{Re}}(s)>1}$

##### 5
${\displaystyle \sum _{n=1}^{\infty }{\frac {\tau ^{2}(n)}{n^{s}}}={\frac {\zeta ^{4}(s)}{\zeta (2s)}}\qquad {\text{Re}}(s)>1}$

##### 6
${\displaystyle \sum _{n=1}^{\infty }{\frac {\tau (n^{2})}{n^{s}}}={\frac {\zeta ^{3}(s)}{\zeta (2s)}}\qquad {\text{Re}}(s)>1}$

##### 7
${\displaystyle \sum _{n=1}^{\infty }{\frac {\mu ^{2}(n)}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}}\qquad {\text{Re}}(s)>1}$

##### 8
${\displaystyle \sum _{n=1}^{\infty }{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta ^{2}(s)}{\zeta (2s)}}\qquad {\text{Re}}(s)>1}$

##### 9
${\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}}\qquad {\text{Re}}(s)>1}$

##### 10
${\displaystyle \sum _{n=1}^{\infty }{\frac {\tau (n)}{n^{s}}}=\zeta ^{2}(s)\qquad {\text{Re}}(s)>1}$

##### 11
${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma (n)}{n^{s}}}=\zeta (s)\,\zeta (s-1)\qquad {\text{Re}}(s)>2}$

##### 12
${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\,\zeta (s-a)\qquad {\text{Re}}(s)>a+1}$

##### 13
${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\,\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\,\zeta (s-a)\,\zeta (s-b)\,\zeta (s-a-b)}{\zeta (2s-a-b)}}}$