DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Induced Dirichlet characters

Definition

Let $d ∣ q$ , and let $χ^{*}$ be a Dirichlet character modulo $d$ . Its induction to modulus $q$ is the Dirichlet character $χ$ modulo $q$ whose arithmetic function is $χ (n) = {\begin{cases} χ^{*} (n), & (n, q) = 1, \\ 0, & (n, q) > 1. \end{cases}$ Equivalently, on units modulo $q$ it is the pullback along reduction to units modulo $d$ . Thus it can be zero at an integer which is a unit modulo $d$ but not modulo $q$ ; this is stronger than merely composing unit-group maps.

Depends on

Dirichlet characters modulo q

Used by

A modulus need not be a conductor Counterexample
Primitive Dirichlet characters and conductor Definition
Unique primitive ancestor of a Dirichlet character Theorem

Dependency tree · one level

1 result 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

Nickolas Andersen, Analytic Number Theory, Lemma 16.1 (standard reference, not scraped)