Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes
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arXiv
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| Natura: | Preprint |
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2024
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| author | Hilberdink, Titus Pushnitski, Alexander |
| author_facet | Hilberdink, Titus Pushnitski, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06892 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes Hilberdink, Titus Pushnitski, Alexander Spectral Theory 11C20 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. |
| title | Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes |
| topic | Spectral Theory 11C20 |
| url | https://arxiv.org/abs/2401.06892 |