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

Sources

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• 1915: G.H. Hardy and Marcel Riesz: The General Theory of Dirichlet's Series ... (next): $\text {II}$: Elementary Theory of the Convergence of Dirichlet's series $\S 3$