uniqueness of Moebius function

Here is a sample result for the function, essentially classifying its uniqueness :

PropositionPlanetmathPlanetmathPlanetmath 1: μ is the unique mapping * such that

μ(1) = 1 (1)
d|nμ(d) = 0 for all n>1 (2)

Proof: By inductionMathworldPlanetmath, there can only be one function with these properties. μ clearly satisfies (1), so take some n>1. Let p be some prime factorMathworldPlanetmath of n, and let m be the productPlanetmathPlanetmath of all the prime factors of n.

d|nμ(d) = d|mμ(d)
= d|mpdμ(d)+d|mpdμ(d)
= d|m/pμ(d)-d|m/pμ(d)
= 0
Title uniqueness of Moebius function
Canonical name UniquenessOfMoebiusFunction
Date of creation 2013-03-22 14:17:25
Last modified on 2013-03-22 14:17:25
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 11A25