ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-07

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.

Perron inversion for a finite Dirichlet polynomial

Example

For $F (s) = 1 + 2^{- s}$ , Perron's integral is $1$ for $1 < x < 2$ , $3 / 2$ for $x = 2$ , and $2$ for $x > 2$ .

Verification

Given: the Perron formula for finite sums.

1.1

Termwise inversion gives $δ (x) + δ (x / 2)$ ; for $x > 1$ , the first term is $1$ .

givenalgebra

2.1

The second term is $0, 1 / 2, 1$ according as $x < 2, x = 2, x > 2$ , proving all three values without a limit interchange.

step 1.1cases∎

Depends on

Perron's inversion formula

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

Kiran S. Kedlaya, Analytic Number Theory, §10.1 (standard reference, not scraped)