Inverse divisor functions using Dirichlet convolution

The study of convolution sums of restricted divisor functions requires more complex computations because the coefficients in the cusp form or the new form are used. In this article, we introduce the inverse divisor function as a tool to give relatively simple formulae for restricted divisor functions using the Dirichlet convolution and show its values in simple cases. The basic properties of the arithmetic functions needed to obtain the main results are introduced.