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We consider the family of arithmetical matrices given explicitly by$$E=\left\{\frac{[n,m]^t}{(nm)^{(\rho+t)/2}}\right\}_{n,m=1}^\infty$$ where$[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters$\rho$ and $t$ satisfy $t>0$, $\rho>t+1$. We prove that $E$ is a compactself-adjoint operator on $\ell^2(\mathbb N)$ with infinitely many of bothpositive and negative eigenvalues. Furthermore, we prove that the orderedsequence of positive eigenvalues of $E$ obeys the asymptotic relation$$\lambda^+_n(E)=\frac{\varkappa}{n^{\rho-t}}(1+o(1)), \quad n\to\infty,$$ withsome $\varkappa>0$ and the negative eigenvalues obey the same relation, withthe same asymptotic coefficient $\varkappa$. We also indicate a connection ofthe spectral analysis of $E$ to the theory of Beurling primes.

Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes