Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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.

Nonprincipal Dirichlet L-functions do not vanish on Re s = 1 away from s = 1

Statement

If χ≠χ0 is a Dirichlet character, then L(1+it,χ)≠0 for every real t≠0.

Facts & Assumptions

Given: A nonprincipal Dirichlet character χ and a real number t≠0.

[L1]

The full product ∏ψ mod qL(s,ψ) has no zero on the line Re⁡s=1 (The full product of Dirichlet L-functions has no zero on Re s = 1).

Proof

technique · direct
1.1

Suppose L(1+it,χ)=0. Then the finite product over all characters modulo q also vanishes at 1+it, because one factor is zero there.

L1givenassume-contra
2.1

This contradicts [L1]. Therefore L(1+it,χ)≠0 for every real t≠0.

step 1.1L1discharge-contradiction∎

Depends on

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