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 $Re s > 1$ , $L (s, χ) = L (s, χ^{*}) \prod_{p ∣ q, p ∤ d} (1 - χ^{*} (p) p^{- s}) .$

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

1.1

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 $(1 - χ^{*} (p) p^{- s})^{- 1}$ .

F1F2given

2.1

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

step 1.1algebra∎

Depends on

Used by

Finite Euler factor from modulus 4 to 12 Example

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

Nickolas Andersen, Analytic Number Theory, equation (16.2) (standard reference, not scraped)