Definition:Abscissa of Convergence

Definition
Let $a_n : \N \to \C$ be an arithmetic function.

Let $F(s)$ be its Dirichlet series.

Let $C \subset \C$ be the set of complex numbers $s$ such that $F(s)$ converges.

The abscissa of convergence of $F$ is the infimum:
 * $\inf\{\sigma \in \R : \forall t\in \R : \sigma+it \in C \}$

Also see

 * Definition:Abscissa of Absolute Convergence
 * Definition:Radius of Convergence