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 analytic continuation of zeta is not the same object as the defining Dirichlet series outside
Remark
The defining series names zeta only on the half-plane . Outside that domain, the symbol refers to the meromorphic continuation from The Riemann zeta function extends meromorphically to the complex plane with its only pole at , not to a literally convergent sum of the original terms.
The standard cautionary value is from The Riemann zeta function has the standard Bernoulli special values at the positive even and nonpositive integers. This identity belongs to analytic continuation and regularization language. It does not say that the ordinary series converges in the usual sense.
Depends on
Used by
- FALSE: ζ(-1) is the ordinary sum 1+2+3+⋯ False statement
Dependency tree · two levels
12 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 11 §3 (standard reference, not scraped)