Definition:Abscissa of Convergence

Definition
Let $\map f s$ be a Dirichlet series

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


 * $\ds \sigma_0 = \inf \set {\map \Re s: s \in \C, \map f s \text{ converges} }$

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

Also see

 * Existence of Abscissa of Convergence, which shows that:
 * if $\map \Re s < \sigma_0$, then $\map f s$ diverges
 * if $\map \Re s > \sigma_0$, then $\map f s$ converges