Riemann zeta function

黎曼 ζ 函数

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Other names / variants: Riemann zeta

Translation evidence: Literature usage

Meaning in context

The function ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge1}n^{-s} for Re⁡s>1\operatorname{Re}s>1, continued meromorphically to C\mathbb C with a simple pole at 11.

在 Re⁡s>1\operatorname{Re}s>1 由 ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge1}n^{-s} 定义，并亚纯延拓到复平面的函数；在 11 处有一个单极点。

Example

ζ(2)=1+1/4+1/9+⋯=π2/6\zeta(2)=1+1/4+1/9+\cdots=\pi^2/6.

Usage and distinctions

The defining series does not converge throughout the continuation domain; zeta at a negative argument means analytic continuation.

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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 specialized to trivial character · 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 标题及定义4.4后的 黎曼ζ函数，printed page 103 · Checked 2026-10-14
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