Definition:Abscissa of Convergence

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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


Sources