A characterization of a certain class of arithmetical multiplicative functions

Abstract

The object of this paper is the set of the "arithmetical multiplicative functions", i.e. the functions ℕ --> ℂ for which f(mn)=f(n)f(m), under the condition that m and n have no common factors. This set is a group with respect to the Dirichlet's convolution. We define for such functions the concept of type (briefly, a number that expresses the fact that f(pn) is zero when n is large enough); moreover, we prove that the set of the completely multiplicative functions which do not assume the value zero coincides with the set of the functions whose inverses are of type 1.

Keywords

• Multiplicative Arithmetical Functions
• Möbius Function