ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES

Abstract Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer n≥1, let $S_n=\{x_1, {\ldots}\, x_n\}$. Let r>0 be a real number and q≥ 1 a given integer. Let $\lambda _n^{(1)}\, {\le}\, {\ldots}\, {\le}\, \lambda _n^{(n)}$ be the eigenvalues of the reciprocal power LCM matrix $(\frac{1}{[x_i, x_j]^r})$ having the reciprocal power ${1\over {[x_i, x_j]^r}}$ of the least common multiple of xi and xj as its i, j-entry. We show that the sequence $\{\lambda _n^{(q)}\}_{n=q}^{\infty}$ converges and ${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(q)}=0$. We show that the sequence $\{\lambda _n^{(n-q+1)}\}_{n=q}^{\infty}$ converges if $s_r:=\sum_{i=1}^{\infty}{1\over {x_i^r}}<\infty $ and ${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(n-q+1)}\, {\le}\, s_r$. We show also that if r> 1, then the sequence $\{\lambda _{ln}^{(tn-q+1)}\}_{n=1}^{\infty}$ converges and ${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _{ln}^{(tn-q+1)}=0$, where t and l are given positive integers such that t≤l−1.