Characterization of homomorphisms on a group of arithmetic functions (Q1390187)

Let \(A_1\) be the set of all real-valued arithmetic functions with \(f(1)=1\) and let \(f\bullet g\) be the unitary convolution of arithmetic functions [\textit{E. Cohen}, Math. Z. 74, 66-80 (1960; Zbl 0094.02601)], i.e. \[ f\bullet g (n)=\sum_{\substack{ d| n\\ (d,n/d)=1}} f(d)g(n/d). \] Moreover, let \(S(n)\) be the characteristic function of the set of all squares. The authors consider the convolution \(f\circ g=f\bullet g\bullet S\), observe that \(A_1\) with multiplication \(\circ\) is an abelian group and describe a class of homomorphisms of this group into the group of all arithmetic functions vanishing at \(1\), with usual addition as group operation.