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

Altinisik E.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.430, no.4, pp.1313-1327, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 430 Issue: 4
  • Publication Date: 2009
  • Doi Number: 10.1016/j.laa.2008.10.025
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1313-1327
  • Keywords: GCD matrices, Multiplicative functions, Matrix norms, Eigenvalues, The Riemann zeta function, Euler products, COMMON DIVISOR MATRICES, LCM MATRICES, ASYMPTOTIC-BEHAVIOR, EIGENVALUES
  • Gazi University Affiliated: Yes


Let C = {1,2, ... ,m} and f be a multiplicative function such that (f * mu) (k) > 0 for every positive integer k and the Euler product zeta(f) = Pi(p) (1-1/f((p))) converges. Let (C(f)) = (f(i,j)) be the m x 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 zeta(f). Specializing these bounds for the arithmetical functions f = N(epsilon), J(epsilon) and sigma(epsilon) we examine the asymptotic behavior the smallest eigenvalue of each of matrices (C(N epsilon)), (C(J epsilon)) and (C(sigma epsilon)) depending on epsilon 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]. (C) 2008 Elsevier Inc. All rights reserved.