DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The convergence and absolute-convergence abscissae of a Dirichlet series

Definition

For a Dirichlet series $D (s) = \sum_{n \geq 1} a_{n} n^{- s}$ , define its abscissa of convergence and abscissa of absolute convergence by

$σ_{c} : = \inf {σ \in \overline{R} : D (s) converges for every ℜ s > σ},$

and

$σ_{a} : = \inf {σ \in \overline{R} : D (s) converges absolutely for every ℜ s > σ} .$

The infima are taken in the extended real line of The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined, so the values $\pm \infty$ are allowed.

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kiran S. Kedlaya, Notes on Analytic Number Theory, Definition 2.3 (standard reference, not scraped)
Leonard Tomczak, Analytic Number Theory, Theorems 3.3 and 3.4 (standard reference, not scraped)