ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF GREATEST COMMON DIVISOR MATRICES

Abstract

Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer $n\ge 1$, let $S_n=\{x_1,\ldots,x_n\}$. Let $\varepsilon$ be a real number and $q\ge 1$ a given integer. Let \smash{$\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}$} be the eigenvalues of the power GCD matrix $((x_i, x_j)^{\varepsilon})$ having the power $(x_i,x_j)^{\varepsilon}$ of the greatest common divisor of $x_i$ and $x_j$ as its $i,j$-entry. We give a nontrivial lower bound depending on $x_1$ and $n$ for \smash{$\lambda _n^{(1)}$} if $\varepsilon>0$. Especially for $\varepsilon>1$, this lower bound is given by using the Riemann zeta function. Let $x\ge 1$ be an integer. For a sequence \smash{$\{x_i\}_{i=1}^{\infty }$} satisfying that $(x_i, x_j)=x$ for any $i\ne j$ and \smash{$\sum_{i=1}^{\infty }{1\over {x_i}}=\infty$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(1)}=x_1^{\varepsilon}-x^{\varepsilon }$}. Let $a\ge 0, b\ge 1$ and $e\ge 0$ be any given integers. For the arithmetic progression \smash{$\{x_{i-e+1}=a+bi\}_{i=e}^{\infty}$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(q)}=0$}. Finally, we show that for any sequence \smash{$\{x_i\}_{i=1}^{\infty}$} and any \smash{$\varepsilon>0$, $\lambda_n^{(n-q+1)}$} approaches infinity when $n$ goes to infinity.