Spectral asymptotics for a family of LCM matrices

The family of arithmetical matrices is studied given explicitly by E ( σ , τ 2 &gt; 0 encoding="application/x-tex">\rho ≔\tau -2\sigma &gt;0 , alttext="tau one half"> encoding="application/x-tex">\tau -\sigma &gt;\frac 12 . It proved that right-parenthesis"> encoding="application/x-tex">E(\sigma ) a compact selfadjoint positive definite operator on alttext="script l squared double-struck upper N ℓ<!-- ℓ mathvariant="double-struck">N encoding="application/x-tex">\ell ^2(\mathbb {N}) ordered sequence eigenvalues obeys asymptotic relation alttext="lamda kappa rho plus o negative right-arrow λ<!-- λ </mml:msub> ϰ<!-- ϰ <mml:mo>+ o stretchy="false">→<!-- → \lambda _n(E(\sigma ))=\frac {\varkappa (\sigma )}{n^\rho }+o(n^{-\rho }), \quad n\to \infty with some alttext="kappa encoding="application/x-tex">\varkappa )&gt;0 This fact applied to asymptotics singular values truncated multiplicative Toeplitz symbol Riemann zeta function vertical line abscissa alttext="sigma slash 2"> / encoding="application/x-tex">\sigma &gt;1/2 relationship spectral analysis theory generalized prime systems also pointed out.