Gauss sum of a Dirichlet character

狄利克雷特征的高斯和

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: character Gauss sum

Translation evidence: Literature usage

Meaning in context

For a Dirichlet character $\chi$ modulo $N$, $\tau(\chi)=\sum_{a\bmod N}\chi(a)e^{2\pi i a/N}$.

模 $N$ 的狄利克雷特征 $\chi$ 的高斯和为 $\tau(\chi)=\sum_{a\bmod N}\chi(a)e^{2\pi i a/N}$。

Example

For the nontrivial character modulo $3$, $\tau(\chi)=e^{2\pi i/3}-e^{4\pi i/3}=i\sqrt3$.

Usage and distinctions

The magnitude formula $|\tau(\chi)|=\sqrt N$ requires primitivity; the finite sum is defined more generally.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 4.1.1, equation (4.1), printed page 104; magnitude result follows Lemma 4.5 · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义§4.1.1，printed page 104 高斯和 及公式(4.1) · Checked 2026-10-14

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