ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

The functional equation shows that $ζ (- 2) = 0$ through the sine factor

Example

$ζ (- 2) = 0.$

Facts & Assumptions

Given: The trivial-zero theorem.

[L1]

For every integer $m \geq 1$ , $ζ (- 2 m) = 0$ (The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip).

Verification

technique · direct

1.1

Apply [L1] with $m = 1$ . Then $ζ (- 2) = 0$ .

L1given

2.1

This is exactly the first trivial zero.

step 1.1algebra∎

Depends on

The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Ch. 6 §2.1 (standard reference, not scraped)

The functional equation shows that ζ(−2)=0ζ(−2)=0 through the sine factor