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.
FALSE: the Riemann zeta function is entire
Statement
False claim: the Riemann zeta function is entire.
Facts & Assumptions
Given: The right-half-plane continuation theorem for zeta.
The defining Dirichlet series extends meromorphically to with a simple residue-one pole at (For , zeta admits the fractional-part integral formula with a simple residue-one pole at ).
Refutation
By [L1], any global extension of zeta still has a pole at , because it agrees with this local meromorphic function near . Therefore it is not holomorphic there.
An entire function is holomorphic on all of , so step 1.1 rules that out. Hence zeta is not entire.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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
- Elias M. Stein and Rami Shakarchi, Complex Analysis, Theorem 2.4 (standard reference, not scraped)