On a class of \(\psi\)-convolutions characterized by the identical equation (Q558131)

\textit{R.~Vaidyanathaswamy} [Bull. Am. Math. Soc. 36, 762--772 (1930; JFM 56.0873.03)] termed the equation \[ f(mn) = \sum_{\substack{ a| m\\ b| n}} f(m/a)f(n/b)f^{-1}(ab)C(a, b) \] involving the Dirichlet convolution as the identical equation for multiplicative functions. \textit{V.~Sita\-ramaiah} and \textit{M.V.~Subbarao} [Proc. Am. Math. Soc. 124, No. 2, 361--369 (1996; Zbl 0847.11003) and J. Indian Math. Soc., New Ser. 64 , No. 1--4, 131--150 (1997)] proved that if \(\psi\) is a Lehmer-Narkiewicz convolution, then a \(\psi\)-generalization of the identical equation holds for all multiplicative functions. The present authors show the converse: If \(\psi\) satisfies certain conditions and the \(\psi\)-generalization of the identical equation holds for all multiplicative functions, then \(\psi\) is a Lehmer-Narkiewicz convolution.