Spectral asymptotics for a family of LCM matrices

The family of arithmetical matrices is studied given explicitly by
$$ E(\sigma,\tau)= \Big\{\frac{n^\sigma m^\sigma}{[n,m]^\tau}\Big\}_{n,m=1}^\infty, $$
where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $\sigma$ and $\tau$ satisfy $\rho:=\tau-2\sigma>0$, $\tau-\sigma>\frac12$, and $\tau>0$. It is proved that $E(\sigma,\tau)$ is a compact selfadjoint positive definite operator on $\ell^2(\mathbb{N})$, and the ordered sequence of eigenvalues of $E(\sigma,\tau)$ obeys the asymptotic relation
$$ \lambda_n(E(\sigma,\tau))=\frac{\varkappa(\sigma,\tau)}{n^\rho}+o(n^{-\rho}),\quad n\to\infty, $$
with some $\varkappa(\sigma,\tau)>0$. This fact is applied to the asymptotics of singular values of truncated multiplicative Toeplitz matrices with the symbol given by the Riemann zeta function on the vertical line with abscissa $\sigma<1/2$. The relationship of the spectral analysis of $E(\sigma,\tau)$ with the theory of generalized prime systems is also pointed out.