Some properties of totients

Abstract

A arithmetical function f is said to be a totient if there exist completely multiplicative functions \(f_t\) and \(f_v\) such that\( f=f_t*f_v^{-1}, \) where \(*\) is the Dirichlet convolution. Euler’s \(\phi \)-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.