Let \(\{x_i\}_{i=1}^{\infty}\) be a strictly increasing sequence of positive integers, let \(r\) be a positive real number and let \(q\) be a positive integer. Let \(\lambda_n^{(1)}\leq \cdots\leq\lambda_n^{(n)}\) be the eigenvalues of the reciprocal power LCM matrix \(({1\over [x_{i},x_{j}]^{r}})\). The authors show, among others, that \(\lim_{n\rightarrow\infty}\lambda_n^{(q)}=0\), \(\lim_{n\rightarrow\infty} \lambda_{n}^{(n-q+1)}\leq\sum_{i=1}^{\infty}{1\over x_{i}^{r}}\) and, for \(r>1\), \(\lim_{n\rightarrow\infty}\lambda_{ln}^{(tn-q+1)}=0\), where \(t\) and \(l\) are positive integers such that \(t\leq l-1\).
For a list of papers on eigenvalues of GCD related matrices see P. Ilmonen, P. Haukkanen and J. K. Merikoski [Linear Algebra Appl. 429, No. 4, 859–874 (2008; Zbl 1143.15016)].