Möbius function on positive integers

默比乌斯函数

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

Other names / variants: arithmetic Möbius function

Translation evidence: Literature usage

Meaning in context

$\mu(1)=1$; $\mu(n)=(-1)^r$ if $n$ is a product of $r$ distinct primes, and $\mu(n)=0$ if a prime square divides $n$.

$\mu(1)=1$；若 $n$ 是 $r$ 个不同素数的乘积则 $\mu(n)=(-1)^r$；若有素数平方整除 $n$ 则 $\mu(n)=0$。

Example

$\mu(30)=-1$, while $\mu(12)=0$.

Usage and distinctions

This is the arithmetic function on positive integers; Möbius functions of arbitrary partially ordered sets have a broader domain.

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 1.2.4, footnote 2, printed page 11: definition of mu · 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§1.2.4，printed page 11 脚注2 默比乌斯函数 · Checked 2026-10-14

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