The Inverse of a Completely Multiplicative Function

Theorem 1

The Dirichlet inverse of a completely multiplicative functionis given by the formulafor all

Proof: LetIfis completely multiplicative then

sinceandforhence

Conversely supposeTo show thatis completely multiplicative it suffices to prove thatfor all primesand integersThe equationimpliesfor

Hence takingwe havefrom whichsois completely multiplicative.

Example

Euler's totient function sobutsinceis completely multiplicative soso

Theorem 2

Ifis multiplicative

Proof: Letthenis multiplicative so to determineit suffices to find

Hence