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Extension by zero is well defined and periodic

Statement

Let $χ$ be a Dirichlet character modulo $q$ . Then the zero extension from Dirichlet characters modulo q is independent of the chosen integer representative, is periodic modulo $q$ , and satisfies $χ (n) = 0$ exactly when $(n, q) > 1$ .

Facts & Assumptions

Given: A modulus $q \geq 1$ and a Dirichlet character $χ$ modulo $q$ in the sense of Dirichlet characters modulo q.

[L1]

A Dirichlet character modulo $q$ is a homomorphism $\overset{ˉ}{χ} : (Z / q Z)^{\times} \to C^{\times}$ , extended by zero on nonunits (Dirichlet characters modulo q).

Proof

technique · direct

1.1

If $m \equiv n (m o d q)$ , then $m$ and $n$ determine the same residue class in $Z / q Z$ . Hence $(m, q) = 1$ iff $(n, q) = 1$ , because both conditions say exactly that this common class is a unit. When they are units, [L1] gives $χ (m) = \overset{ˉ}{χ} (\overset{ˉ}{m}) = \overset{ˉ}{χ} (\overset{ˉ}{n}) = χ (n)$ ; when they are nonunits, [L1] gives $χ (m) = χ (n) = 0$ .

L1givenalgebra

2.1

Step 1.1 is exactly representative-independence, and applying it to $m = n + q$ gives $χ (n + q) = χ (n)$ for every integer $n$ , so $χ$ is $q$ -periodic. The final clause of [L1] says $χ (n) = 0$ precisely on the nonunit residue classes, equivalently exactly when $(n, q) > 1$ .

L1step 1.1algebra∎

Depends on

Dirichlet characters modulo q

Used by

Dependency tree · one level

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Sources

Andrew V. Sutherland, Number Theory I, section 18.2 (standard reference, not scraped)
Kiran S. Kedlaya, Notes on Analytic Number Theory, section 3.1 (standard reference, not scraped)