Some properties of totients

Abstract

A arithmetical function f is said to be a totient if there exist completely multiplicative functions ft and fv such thatf=ft∗fv-1, where ∗ is the Dirichlet convolution. Euler’s ϕ-function is an important example of a totient. In this paper we find the structure of the usual product of two totients, the usual integer power of totients, the usual product of a totient and a specially multiplicative function and the usual product of a totient and a completely multiplicative function. These results are derived with the aid of generating series. We also provide some distributive-like characterizations of totients involving the usual product and the Dirichlet convolution of arithmetical functions. They give as corollaries characterizations of completely multiplicative functions.