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This article is cited in 2 scientific papers (total in 2 papers)
Research Papers
Spectral asymptotics for a family of LCM matrices
T. Hilberdinka, A. Pushnitskib a Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading, RG6 6AX, U.K.
b Department of Mathematics, King's College London, Strand, London, WC2R 2LS, U.K.
Abstract:
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.
Keywords:
LCM matrix, arithmetical matrix, multiplicative Toeplitz matrix, eigenvalue asymptotics.
Received: 25.10.2021
Citation:
T. Hilberdink, A. Pushnitski, “Spectral asymptotics for a family of LCM matrices”, Algebra i Analiz, 34:3 (2022), 207–231; St. Petersburg Math. J., 34:3 (2023), 463–481
Linking options:
https://www.mathnet.ru/eng/aa1816 https://www.mathnet.ru/eng/aa/v34/i3/p207
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