Alphabeta Math
ExampleConstruction: AI-adaptedVerification: 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.

Missing Euler factors for a principal Dirichlet L-function

Example

For small moduli, the principal Dirichlet L-function is obtained from zeta by removing exactly the Euler factors at the primes dividing the modulus.

Facts & Assumptions

Given: The principal factorization theorem (The principal Dirichlet L-function factors through zeta).

Verification

technique · direct
1.1

For q=4, the theorem gives L(s,χ0)=ζ(s)(1−2−s). For q=6, it gives L(s,χ0)=ζ(s)(1−2−s)(1−3−s). In each case the omitted Euler factors are exactly those at the bad primes dividing the modulus.

givenalgebra
2.1

Evaluating the finite factor at s=1 gives the residues 1−1/2=1/2=φ(4)/4 and (1−1/2)(1−1/3)=1/3=φ(6)/6, matching the general residue formula.

step 1.1givenalgebra∎

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