Quotients of arithmetical functions under the Dirichlet convolution

Abstract

We study existence of a solution of the arithmetical equation $f\ast h = g$ in $f,$ where $f\ast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $\mu$ and the so-called totients. As applications we also present our results on the arithmetical equation $f\ast h = g$ in the language of Cauchy convolution and further deconvolution in discrete linear systems.