Formelsammlung Mathematik: Unendliche Reihen: Dirichletreihen

Zurück zu Unendliche Reihen

1

$\sum _{n=1}^{\mathrm{\infty }}\frac{\phi (n)}{n^s}=\frac{\zeta (s-1)}{\zeta (s)}\text{Re}(s)>2$

2

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

3

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

4

$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Lambda }(n)}{n^s}=-\frac{{\zeta }^{\prime }(s)}{\zeta (s)}\text{Re}(s)>1$

5

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

6

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

7

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

8

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

9

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

10

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

11

$\sum _{n=1}^{\mathrm{\infty }}\frac{\sigma (n)}{n^s}=\zeta (s)\zeta (s-1)\text{Re}(s)>2$

12

$\sum _{n=1}^{\mathrm{\infty }}\frac{{\sigma }_a(n)}{n^s}=\zeta (s)\zeta (s-a)\text{Re}(s)>a+1$

13

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