ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES

Abstract

AbstractLet$\{x_i\}_{i=1}^{\infty}$be an arbitrary strictly increasing infinite sequence of positive integers. For an integern≥1, let$S_n=\{x_1, {\ldots}\, x_n\}$. Letr>0 be a real number andq≥ 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 ofxiandxjas itsi,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$. We show also that ifr> 1, then the sequence$\{\lambda _{ln}^{(tn-q+1)}\}_{n=1}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _{ln}^{(tn-q+1)}=0$, wheretandlare given positive integers such thatt≤l−1.