DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Dirichlet characters modulo q

Definition

Let $q \geq 1$ . A Dirichlet character modulo $q$ is the datum of a group homomorphism

$\overset{ˉ}{χ} : (Z / q Z)^{\times} \to C^{\times} .$

Its associated arithmetic function is the map $χ : Z \to C$ defined by

$χ (n) = {\begin{cases} \overset{ˉ}{χ} (\overset{ˉ}{n}), & (n, q) = 1, \\ 0, & (n, q) > 1, \end{cases}$

where $\overset{ˉ}{n}$ is the residue class of $n$ modulo $q$ . The modulus is part of the datum: the same arithmetic function may arise from different nonminimal moduli, and that conductor story is not built into this definition.

Used by

Dirichlet L-functions Definition
The principal character modulo q Definition
Dirichlet character tables modulo 3, 4, and 5 Example
Dirichlet character tables modulo 8 and 12 Example
Arithmetic characterization of Dirichlet characters modulo q Lemma
Character values on units are roots of unity Lemma
Extension by zero is well defined and periodic Lemma
Orthogonality relations for Dirichlet characters modulo q Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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