LemmaStatement: 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.

Arithmetic characterization of Dirichlet characters modulo q

Statement

A function $χ : Z \to C$ comes from a Dirichlet character modulo $q$ if and only if all of the following hold:

$χ$ is $q$ -periodic.
$χ (m n) = χ (m) χ (n)$ for all integers $m, n$ .
$χ (n) = 0$ exactly when $(n, q) > 1$ .
$χ (1) = 1$ .

Facts & Assumptions

Given: A positive integer $q$ and a function $χ : Z \to C$ .

[L1]

Every Dirichlet character modulo $q$ is extended by zero from a homomorphism on $(Z / q Z)^{\times}$ (Dirichlet characters modulo q).

[L2]

That extension is representative-independent, $q$ -periodic, and vanishes exactly on the nonunits modulo $q$ (Extension by zero is well defined and periodic).

Proof

technique · direct

1.1

Assume first that $χ$ comes from a Dirichlet character modulo $q$ . Periodicity and the support condition are exactly [L2]. If $(m, q) > 1$ or $(n, q) > 1$ , then both $χ (m n)$ and $χ (m) χ (n)$ are $0$ by [L2]. If both are coprime to $q$ , then [L1] gives $χ (m n) = \overset{ˉ}{χ} (\overset{ˉ}{m} \overset{ˉ}{n}) = \overset{ˉ}{χ} (\overset{ˉ}{m}) \overset{ˉ}{χ} (\overset{ˉ}{n}) = χ (m) χ (n)$ . Also $χ (1) = \overset{ˉ}{χ} (\overset{ˉ}{1}) = 1$ because every homomorphism sends the identity to the identity.

L1L2givenalgebra

1.2

Conversely, assume properties 1-4. If $(n, q) = 1$ , define $\overset{ˉ}{χ} (\overset{ˉ}{n}) : = χ (n)$ . This is well defined because property 1 makes $χ$ constant on residue classes modulo $q$ , and property 3 shows that only unit classes receive nonzero values. For unit classes $\overset{ˉ}{m}, \overset{ˉ}{n}$ , property 2 gives $\overset{ˉ}{χ} (\overset{ˉ}{m} \overset{ˉ}{n}) = χ (m n) = χ (m) χ (n) = \overset{ˉ}{χ} (\overset{ˉ}{m}) \overset{ˉ}{χ} (\overset{ˉ}{n})$ , so $\overset{ˉ}{χ}$ is a homomorphism $(Z / q Z)^{\times} \to C^{\times}$ ; property 4 makes it unital. Extending this homomorphism by zero recovers the original $χ$ by property 3.

givenalgebra

2.1

Step 1.1 proves necessity and step 1.2 proves sufficiency.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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

Andrew V. Sutherland, Number Theory I, Definition 18.4 and Definition 18.6 (standard reference, not scraped)
Kiran S. Kedlaya, Notes on Analytic Number Theory, section 3.1 (standard reference, not scraped)