ExampleConstruction: AI-adaptedVerification: 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.
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An orthogonality table for Dirichlet characters

Example

Modulo $5$ , the character table from Dirichlet character tables modulo 3, 4, and 5 verifies both orthogonality relations and the residue-class indicator numerically.

Facts & Assumptions

Given: The four characters modulo $5$ , the orthogonality theorem, and the indicator corollary (Dirichlet character tables modulo 3, 4, and 5, Orthogonality relations for Dirichlet characters modulo q, A residue-class indicator from character sums).

Verification

technique · direct

1.1

Using the four rows $(1, 1, 1, 1)$ , $(1, i, - i, - 1)$ , $(1, - 1, - 1, 1)$ , and $(1, - i, i, - 1)$ on the unit classes $1, 2, 3, 4$ , the row inner products are $4$ on the diagonal and $0$ off the diagonal. Likewise the column sums $\sum_{χ} χ (a) \overline{χ (b)}$ are $4$ when $a = b$ and $0$ otherwise. This is the theorem Orthogonality relations for Dirichlet characters modulo q in the concrete case $q = 5$ .

givenalgebra

2.1

Taking $a = 2$ , the weighted average $\frac{1}{4} \sum_{χ} \overline{χ (2)} χ (n)$ is $1$ at the class $2$ and $0$ at the other residue classes, exactly as A residue-class indicator from character sums predicts.

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Leonard Tomczak, Analytic Number Theory, Corollary 4.3 (standard reference, not scraped)