On the eigenvalues of certain number-theoretic matrices

TL;DR

This paper investigates the eigenvalues of a class of number-theoretic matrices with entries based on gcd and lcm, providing bounds and asymptotic behavior analysis for their eigenvalues.

Contribution

It offers new bounds and insights into the eigenvalues of matrices defined by gcd and lcm, extending current understanding beyond existing O-estimates.

Findings

Derived bounds for eigenvalues of the matrices.

Analyzed asymptotic behavior when nd eta are equal.

Extended knowledge beyond existing O-estimates.

Abstract

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