# Definition:Abscissa of Convergence

## Definition

Let $f \left({s}\right)$ be a Dirichlet series

The abscissa of convergence of $f$ is the extended real number $\sigma_0 \in \overline \R$ defined by:

$\sigma_0 = \displaystyle \inf \left\{ {\operatorname{Re} \left({s}\right) : s \in \C, f \left({s}\right) \text{converges} }\right\}$

where $\inf \varnothing = +\infty$.

## Also see

if $\operatorname{Re} \left({s}\right) < \sigma_0$, then $f \left({s}\right)$ diverges
if $\operatorname{Re} \left({s}\right) > \sigma_0$, then $f \left({s}\right)$ converges