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

TL;DR

This work analyzes an infinite arithmetical matrix $E$ with entries $E_{n,m}=\frac{[n,m]^t}{(nm)^{(\rho+t)/2}}$, under $t>0$ and $\rho>t+1$, showing that $E$ is a compact self-adjoint operator on $\ell^2(\mathbb N)$ with infinitely many positive and negative eigenvalues. The authors establish precise eigenvalue asymptotics: the positive and negative eigenvalues satisfy $\lambda^+_n(E)=\frac{\varkappa}{n^{\rho-t}}(1+o(1))$ and $\lambda^-_n(E)=\frac{\varkappa}{n^{\rho-t}}(1+o(1))$, for some $\varkappa>0$, revealing an asymptotic symmetry in the spectrum. The proof develops a detailed analysis of the tensor-product-like structure $E\approx\bigotimes_p E_p$ over primes, using spectral perturbation theory and interlacing, then applies a Wiener–Ikehara Tauberian argument to translate spectral data into eigenvalue counting asymptotics. A Beurling prime-system perspective is provided, linking the eigenvalue distribution to generalized prime systems and enabling refined error terms via Beurling zeta connections. Overall, the paper extends the spectral theory of arithmetical matrices to the sign-indefinite, negative-$\tau$ regime and highlights deep ties to Beurling primes, with explicit constants and a clear asymptotic framework for eigenvalue distributions.

Abstract

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

Theorems & Definitions (1)

• proof