Definition:Dirichlet Inverse of Arithmetic Function

Definition
Let $f : \N \to \C$ be an arithmetic function.

Let $\varepsilon$ be the identity arithmetic function.

A Dirichlet inverse of $f$ is an arithmetic function $g$ such that $g*f = \varepsilon$, where $*$ denotes Dirichlet convolution.

That is, an inverse of $f$ in the ring of arithmetic functions.

Also see

 * Units of Ring of Arithmetic Functions