Gcd-closed sets and determinants of matrices associated with arithmetical functions (Q2773338)

Let \(f\), \(g\) and \(h\) be arithmetical functions, and let \(\psi\) denote the arithmetical function of two variables defined as \(\psi(t, r)=\sum_{d|(t, r)} f(d)g(t/d)h(r/d)\). Let \(S=\{x_1, x_2,\ldots, x_n\}\) be a gcd-closed set of positive integers. The author presents a formula for \(\det[\psi(x_i, x_j)]\) under a condition on \(g\) and \(h\). This formula is a slight generalization of a formula in \textit{S. Hong} [Linear Multilinear Algebra 45, No. 4, 349-358 (1999; Zbl 0932.11018)]. Some interesting applications on determinants of matrices associated with classes of arithmetical functions are given. The paper contains some minor misprints.