Dedekind zeta function

戴德金 ζ 函数

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Translation evidence: Literature usage

Meaning in context

For a number field $K$, $\zeta_K(s)=\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}|\mathcal O_K/\mathfrak a|^{-s}$ for $\operatorname{Re}s>1$, with its meromorphic continuation.

数域 $K$ 的戴德金ζ函数在 $\operatorname{Re}s>1$ 定义为 $\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}|\mathcal O_K/\mathfrak a|^{-s}$，并取其亚纯延拓。

Example

For $K=\mathbb Q$, $\zeta_K(s)=\zeta(s)$ since nonzero ideals of $\mathbb Z$ are $n\mathbb Z$.

Usage and distinctions

The sum is over integral ideals rather than algebraic integers; counting elements would repeatedly count generators differing by units.

Read the encyclopedia explanation

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 4.2 opening, printed page 109, and Corollary 4.23, printed page 112, for meromorphic 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§4.2 开头 戴德金ζ函数，printed page 109 · Checked 2026-10-14

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