Rearick's isomorphism and a characterization of
\(\psi\)-additive functions

Abstract. Let \({\Bbb A}_{1}\) denote the set of arithmetic functions \(f\) with \(f(1)\) real and \(P'\subseteq {\Bbb A}_{1}\) denote the set of arithmetic functions \(f\) with \(f(1)>0\). If \(\psi\) denotes Lehmer's convolution, placing mild conditions on \(\psi\) it can be shown that the \(\psi\)-analogue of Rearick's ([7]) logarithmic operator \(L: (P',\psi) \to ({\Bbb A}_{1},+)\) defined by \(Lf(1)=\log f(1)\) and \(Lf (n)=(fh\psi f^{-1})(n)\), if \(n>1\), where \(h\) is any \(\psi\)-additive function with \(h(n)\neq 0\) for all \(n>1\), is a group isomorphism. In this paper we prove the converse when \(\psi\) is a Lehmer–Narkiewicz convolution.