Spectral asymptotics for a family of LCM matrices

Abstract

The family of arithmetical matrices is studied given explicitly by E ( σ , τ ) = { n σ m σ [ n , m ] τ } n , m = 1 ∞ , \begin{equation*} E(\sigma ,\tau )= \bigg \{\frac {n^\sigma m^\sigma }{[n,m]^\tau }\bigg \}_{n,m=1}^\infty , \end{equation*} where [ n , m ] [n,m] is the least common multiple of n n and m m and the real parameters σ \sigma and τ \tau satisfy ρ ≔ τ − 2 σ > 0 \rho ≔\tau -2\sigma >0 , τ − σ > 1 2 \tau -\sigma >\frac 12 , and τ > 0 \tau >0 . It is proved that E ( σ , τ ) E(\sigma ,\tau ) is a compact selfadjoint positive definite operator on ℓ 2 ( N ) \ell ^2(\mathbb {N}) , and the ordered sequence of eigenvalues of E ( σ , τ ) E(\sigma ,\tau ) obeys the asymptotic relation λ n ( E ( σ , τ ) ) = ϰ ( σ , τ ) n ρ + o ( n − ρ ) , n → ∞ , \begin{equation*} \lambda _n(E(\sigma ,\tau ))=\frac {\varkappa (\sigma ,\tau )}{n^\rho }+o(n^{-\rho }), \quad n\to \infty , \end{equation*} with some ϰ ( σ , τ ) > 0 \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 σ > 1 / 2 \sigma >1/2 . The relationship of the spectral analysis of E ( σ , τ ) E(\sigma ,\tau ) with the theory of generalized prime systems is also pointed out.