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.

Symmetric finite zero products model the genus-one product for xi

Example

Fix a complex number $ρ$ with $ρ \notin R$ and consider the finite product

$P_{ρ} (s) : = (1 - \frac{s}{ρ}) (1 - \frac{s}{\overline{ρ}}) (1 - \frac{s}{1 - ρ}) (1 - \frac{s}{1 - \overline{ρ}}) .$

Then $P_{ρ}$ has real coefficients and satisfies $P_{ρ} (1 - s) = P_{ρ} (s)$ . This is the polynomial shadow of the full xi product.

Facts & Assumptions

Given: The zero symmetries behind the xi Hadamard product.

[L1]

The xi function has a genus-one canonical product over the nontrivial zeros of zeta (The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta).

Verification

technique · direct

1.1

The four listed roots of $P_{ρ}$ , counted with multiplicity, are closed under complex conjugation and under $α \mapsto 1 - α$ . Therefore the coefficients of $P_{ρ}$ are real, and replacing $s$ by $1 - s$ permutes this root multiset.

L1givenalgebra

2.1

A polynomial is determined by its roots together with the leading coefficient, and both remain unchanged in step 1.1. Hence $P_{ρ} (1 - s) = P_{ρ} (s)$ . This finite symmetric zero set is exactly the pattern that the full xi product repeats over all nontrivial zeros.

step 1.1algebra∎

Depends on

The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)