Extended arithmetic functions (Q2174294)

The authors consider extended arithmetic functions, which are functions, whose domain is the set of positive integers and codomain is a unital associative algebra \(\mathcal{A}\). They investgate multiplicativity and convolutional properties of such functions. The authors also introduce the concept of an arithmetic system of (orthogonal) idempotents of \(\mathcal{A}\) and present number theoretic properties for such systems, e.g., relating to Ramanujan's sum. As an application, the Euler differential operator is represented in terms of the Euler totient function.