Dirichlet series

Definition

Suppose $f:\mathbb{N}\to \mathbb{C}$ is a function. In other words, $f$ is an arithmetic function. The Dirichlet series of $f$ is defined as the following formal series:

$s\mapsto {\sum }_{n\in \mathbb{N}}\frac{f(n)}{n^s}$.

The corresponding function is defined as the sum of the series where convergence of the summation is considered in $\mathbb{C}$. Even though the series may converge on only a part of $\mathbb{C}$, it might be possible to extend to a global meromorphic function, or to a meromorphic function on a large part of $\mathbb{C}$.