Spectral asymptotics for a family of LCM matrices

Abstract:

The family of arithmetical matrices is studied given explicitly by \begin{equation*} E(\sigma ,\tau )= \bigg \{\frac {n^\sigma m^\sigma }{[n,m]^\tau }\bigg \}_{n,m=1}^\infty , \end{equation*} where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $\sigma$ and $\tau$ satisfy $\rho ≔\tau -2\sigma >0$, $\tau -\sigma >\frac 12$, 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 \begin{equation*} \lambda _n(E(\sigma ,\tau ))=\frac {\varkappa (\sigma ,\tau )}{n^\rho }+o(n^{-\rho }), \quad n\to \infty , \end{equation*} 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.