alternate proof of Möbius inversion formula

The Möbius inversion theorem can also be proved elegantly using the fact that arithmetic functionsMathworldPlanetmath form a ring under + and *.

Let I be the arithmetic function that is everywhere 1. Then obviously if μ is the Möbius functionMathworldPlanetmath,


and thus I*μ=e, where e is the identity of the ring.

But then


and so f=g*I. Thus f*μ=g*I*μ=g. But g=f*μ means precisely that


and we are done.

The reverse equivalence is similar (f*μ=gf*μ*I=g*If=g*I).

Title alternate proof of Möbius inversion formulaMathworldPlanetmathPlanetmath
Canonical name AlternateProofOfMobiusInversionFormula
Date of creation 2013-03-22 16:30:31
Last modified on 2013-03-22 16:30:31
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Proof
Classification msc 11A25