Möbius inversion for divisor sums

默比乌斯反演

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

Other names / variants: divisor-sum Möbius inversion

Translation evidence: Literature usage

Meaning in context

For functions on positive integers, F(n)=∑d∣nf(d)F(n)=\sum_{d\mid n}f(d) implies f(n)=∑d∣nμ(n/d)F(d)f(n)=\sum_{d\mid n}\mu(n/d)F(d).

对正整数上的函数，若 F(n)=∑d∣nf(d)F(n)=\sum_{d\mid n}f(d)，则 f(n)=∑d∣nμ(n/d)F(d)f(n)=\sum_{d\mid n}\mu(n/d)F(d)。

Example

From n=∑d∣nφ(d)n=\sum_{d\mid n}\varphi(d) one gets φ(6)=6−3−2+1=2\varphi(6)=6-3-2+1=2.

Usage and distinctions

This divisor-poset formula uses the arithmetic Möbius function; it is not matrix inversion or a Möbius transformation.

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: divisor-sum inversion formula · 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 默比乌斯反演 · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.