Some Perpendicularities of Arithmetical Functions

Abstract

We introduce the notion of perpendicularity of arithmetical functions and discuss a few concrete examples of perpendicularities. Further, we show that the set of arithmetical functions f with f(1) = 1 forms a real vector space with the Dirichlet convolution as addition and real power under the Dirichlet convolution as scalar multiplication. Moreover, we prove that multiplicative functions are perpendicular to antimultiplicative functions with respect to the natural perpendicularity.