On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong-Loewy conjecture

Let C={1,2,…,m} and f be a multiplicative function such that (f∗μ)(k)>0 for every positive integer k and the Euler product converges. Let (C_f)=(f(i,j)) be the m×m matrix defined on the set C having f evaluated at the greatest common divisor (i,j) of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of (C_f)^-1, the inverse of (C_f), in terms of ζ_f. Specializing these bounds for the arithmetical functions f=N^ε,J_ε and σ_ε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (C_N^ε),(C_Jε) and (C_σε) depending on ε when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551-569].