Euler product

The infinite product

$$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$

where $s$ is a real number and $p$ runs through all prime numbers. This product converges absolutely for all $s>1$. The analogous product for complex numbers $s=\sigma+it$ converges absolutely for $\sigma>1$ and defines in this domain the Riemann zeta-function

$$\zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}.$$

Comments[edit]

See also Euler identity and Zeta-function.