Divisibility of arithmetic-function GCD and LCM matrices

Characterize the gcd-closed sets S and multiplicative arithmetic functions f belonging to the class C_S, for which the nonsingular matrix (f(S)) divides the matrix (f[S]) in the ring M_n(ℤ).

Background

The paper studies divisibility relations between power GCD and power LCM matrices on gcd-closed sets, proving a complete characterization for power arithmetic functions. In the final section, the authors broaden the setting to matrices generated by an arbitrary arithmetic function f, defining (f(S)) with entries f((x_i,x_j)) and (f[S]) with entries f([x_i,x_j]).

The class C_S consists of arithmetic functions f for which (f*μ)(d) is an integer whenever d divides lcm(S). The problem asks for a characterization of both the gcd-closed set S and a multiplicative function f∈C_S, under the additional assumption that (f(S)) is nonsingular, that guarantees divisibility of (f[S]) by (f(S)) over the integer matrix ring. The paper notes that its main theorem and earlier results resolve this question when f is a power arithmetic function.