LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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A nonprincipal character has zero complete sum

Statement

Let $χ$ be a Dirichlet character modulo $q$ with $χ \neq χ_{0}$ . Then for every complete residue system $R$ modulo $q$ ,

$\sum_{r \in R} χ (r) = 0.$

Facts & Assumptions

Given: A nonprincipal Dirichlet character $χ$ modulo $q$ .

[L1]

The principal character is $1$ on integers coprime to $q$ and $0$ otherwise (The principal character modulo q).

[L2]

If $(a, q) = 1$ , then $χ (a)$ is a root of unity and hence may differ from $1$ only as a nonzero scalar (Character values on units are roots of unity).

Proof

technique · direct

1.1L1L2givenchoose

Since $χ \neq χ_{0}$ , [L1] shows that some unit $a$ modulo $q$ satisfies $χ (a) \neq 1$ . Let $S : = \sum_{r \in R} χ (r)$ . Multiplication by the unit $a$ permutes the residue classes modulo $q$ , so $a R$ is again a complete residue system modulo $q$ .

2.1step 1.1L2algebra∎

Reindex over $a R$ and use multiplicativity on units: $S = \sum_{r \in R} χ (a r) = χ (a) \sum_{r \in R} χ (r) = χ (a) S$ . Because $χ (a) \neq 1$ , this gives $(χ (a) - 1) S = 0$ , hence $S = 0$ .

Depends on

Used by

Nonprincipal Dirichlet character partial sums are bounded Lemma

Dependency tree · two levels

4 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

Kiran S. Kedlaya, Notes on Analytic Number Theory, Lemma 3.2 (standard reference, not scraped)
Andrew V. Sutherland, Number Theory I, Lemma 18.12 (standard reference, not scraped)