On the eigenvalues of certain number-theoretic matrices

In this paper we study the structure and give bounds for the eigenvalues of the $n\times n$ matrix, which $ij$ entry is $(i,j)^α[i,j]^β$, where $α,β\in\Rset$, $(i,j)$ 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 eigenvalue is known in case when $α=β$.

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