ON THE EIGENVALUES OF CERTAIN NUMBER-THEORETIC MATRICES

Abstract

In this paper we study the structure and give bounds for the eigenvalues of the n×n matrix, whose ij entry is (i,j)α[i,j]β, whereα,β ∈ R, (i,j) is the greatest common divisor of i and j and [i,j] is the least common multiple of i and j. Currently, only O-estimates for the greatest eigenvalue of this matrix can be found in the literature, and the asymptotic behaviour of the greatest and smallest eigenvalues is known in case when α = β.