Alphabeta MathHow 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.
The Riemann xi function
Definition
Assume countable choice.
The completed function extends meromorphically with simple poles at and by The completed zeta function satisfies . The Riemann xi function is defined on by
The role of the factor is to cancel the two simple poles of the completed function . Thus is the entire completion, while remains meromorphic.
Depends on
Used by
- A local formula for the logarithmic derivative of zeta Lemma
- Zeta logarithmic derivative zero bound Lemma
- The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta Theorem
- The Riemann xi function is entire of order one, real on the real axis, and symmetric under s↦1-s Theorem
- The Riemann--von Mangoldt zero count Theorem
Dependency tree · two levels
13 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
- K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 13 §8 (standard reference, not scraped)