 If is an arithmetic function possessing a Dirichlet inverse , then if one defines

then

The previous formula arises in the special case of the constant function , whose Dirichlet inverse is .A particular application of the first of these extensions arises if we have (complex-valued) functions and defined on the positive integers, with

By defining and , we deduce that

A simple example of the use of this formula is counting the number of reduced fractions , where and are coprime and .If we let be this number, then is the total number of fractions with , where and are not necessarily coprime.