Dirichlet L-function

狄利克雷 L 函数

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Other names / variants: Dirichlet L-series

Translation evidence: Literature usage

Meaning in context

For a Dirichlet character χ\chi, L(s,χ)=∑n≥1χ(n)n−sL(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s} for Re⁡s>1\operatorname{Re}s>1, with its meromorphic continuation.

对狄利克雷特征 χ\chi，在 Re⁡s>1\operatorname{Re}s>1 定义 L(s,χ)=∑n≥1χ(n)n−sL(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s}，并取其亚纯延拓。

Example

For the nontrivial character modulo 44, L(2,χ)=1−3−2+5−2−7−2+⋯L(2,\chi)=1-3^{-2}+5^{-2}-7^{-2}+\cdots.

Usage and distinctions

A principal character at modulus N gives zeta with the Euler factors at primes dividing N removed; it equals zeta itself only at modulus 1.

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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Definition 4.4, printed page 103, and Theorem 4.10 on continuation · 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定义4.4 狄利克雷特征的L函数，§4.1标题，printed page 103 · Checked 2026-10-14
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