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 (Cf)=(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 (Cf)-1, the inverse of (Cf), in terms of ζf. Specializing these bounds for the arithmetical functions f=Nε,Jε and σε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (CNε),(CJε) and (Cσε) 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].