Functional equations characterizing totients in regular convolution rings

Abstract

Let A be a regular convolution. An arithmetical function f is A-multiplicative if f(1)=1 and f(mn)=f(m)f(n) for all positive integers m, n with m,n∈A(mn), and f is an A-totient if there exist A-multiplicative functions fT and fV such that f=fT∗AfV-1. In this paper, we present various functional equations characterizing A-totients, derived from fundamental properties of Euler’s totient function. Our exposition also provides a historical perspective.