completely multiplicative functions whose convolution inverses are completely multiplicative

Corollary 1.

The only completely multiplicative function whose convolution inverse is also completely multiplicative is $\epsilon$, the convolution identity function.

Proof.

Let $f$ be a completely multiplicative function whose convolution inverse is completely multiplicative. By this entry (http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction), $f\mu$ is the convolution inverse of $f$, where $\mu$ denotes the Möbius function. Thus, $f\mu$ is completely multiplicative.

Let $p$ be any prime. Then

Thus, $f(p)=0$ for every prime $p$. Since $f$ is completely multiplicative,

Hence, $f=\epsilon$.∎