Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Finite Euler factors under character induction

Statement

If χ modulo q is induced by its primitive ancestor χ∗ modulo d, then, for Re⁡s>1, L(s,χ)=L(s,χ∗)∏p∣q, p∤d(1−χ∗(p)p−s).

Facts & Assumptions

Given: The indicated induced pair and Re⁡s>1.

[F1]

The ancestor has conductor dividing q (Unique primitive ancestor of a Dirichlet character).

[F2]

Dirichlet L-functions have their absolutely convergent Euler products in this half-plane (Euler product for Dirichlet L-functions).

Proof

technique · direct
1.1F1F2given

In [F2], the local factors of χ and χ∗ agree unless p∣q but p∤d; at exactly those primes χ(p)=0, while the primitive factor is (1−χ∗(p)p−s)−1.

2.1step 1.1algebra∎

Cancelling all common local factors and multiplying the finite exceptional set gives the displayed identity.

Depends on

Used by

Dependency tree · two levels

9 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