TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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Norm of a primitive Gauss sum

Statement

For a primitive character $χ$ modulo $q$ , $∣ τ (χ) ∣^{2} = q$ .

Facts & Assumptions

Given: A primitive character $χ$ modulo $q$ .

[F1]

The primitive twist identity holds for every integer (Primitive Gauss-sum twist).

Proof

technique · direct

1.1F1givenalgebra

By the definition of the Gauss sum, $\overline{τ (χ)} = \sum_{a m o d q} \overline{χ (a)} e (- a / q) .$ Multiplying by $τ (χ)$ and using [F1] with $m = a$ gives $∣ τ (χ) ∣^{2} = \sum_{a m o d q} \sum_{b m o d q} χ (b) e (a b / q) e (- a / q) .$

2.1step 1.1algebra∎

The inner sum over $a$ is $q$ when $b \equiv 1 (m o d q)$ and $0$ otherwise. Thus only $b = 1$ remains, and the expression in step 1.1 is $q χ (1) = q$ .

Depends on

Used by

Unit modulus of the Dirichlet root number Corollary
Gauss sum for the character modulo 4 Example
Functional equation for primitive Dirichlet L-functions Theorem

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

Nickolas Andersen, Analytic Number Theory, Lemma 16.4 (standard reference, not scraped)