Note on the functional equationS[(m, n)F[n/(m, n)]=F(n)h[n/(m, n)

Abstract

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σd|n ƒ(d)h(n/d) andS(m n) = Σd|(m, n) ƒ(d)h(n/d). The well-known Holder's identity is a special case of this functional equation (ƒ(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.