Alphabeta MathDefinitionDefinition: 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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Dirichlet series
Definition
A Dirichlet series is a series of the form
where , each , and
uses the real logarithm of the positive integer .
Depends on
Used by
- The convergence and absolute-convergence abscissae of a Dirichlet series Definition
- A Dirichlet series with absolute convergence on a right half-plane is determined there by its coefficients Theorem
- Absolute convergence at one point forces absolute and locally uniform convergence on closed half-planes to the right Theorem
- Convergence at one point of a Dirichlet series forces local uniform convergence on the open half-plane to its right Theorem
- Dirichlet series from arithmetic functions admit the Abel-summation integral formula Theorem
- Multiplying absolutely convergent Dirichlet series gives Dirichlet convolution Theorem
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 2.1 (standard reference, not scraped)
- Leonard Tomczak, Analytic Number Theory, Chapter 3 (standard reference, not scraped)