ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES

Let$\{x_i\}_{i=1}^{\infty}$be an arbitrary strictly increasing infinite sequence of positive integers. For an integern≥1, let$S_n=\{x_1, {\ldots}\, x_n\}$. Letr>0 be a real number andq≥ 1 a given integer. Let$\lambda _n^{(1)}\, {\le}\, {\ldots}\, {\le}\, \lambda _n^{(n)}$be the eigenvalues of the reciprocal power LCM matrix$(\frac{1}{[x_i, x_j]^r})$having the reciprocal power${1\over {[x_i, x_j]^r}}$of the least common multiple ofxiandxjas itsi,j-entry. We show that the sequence$\{\lambda _n^{(q)}\}_{n=q}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(q)}=0$. We show that the sequence$\{\lambda _n^{(n-q+1)}\}_{n=q}^{\infty}$converges if$s_r:=\sum_{i=1}^{\infty}{1\over {x_i^r}}<\infty $and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(n-q+1)}\, {\le}\, s_r$. We show also that ifr> 1, then the sequence$\{\lambda _{ln}^{(tn-q+1)}\}_{n=1}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _{ln}^{(tn-q+1)}=0$, wheretandlare given positive integers such thatt≤l−1.

On Nonsingular Power LCM Matrices

Let e ≥ 1 be an integer and S={x1,…,xn} a set of n distinct positive integers. The matrix ([xi, xj]e) having the power [xi, xj]e of the least common multiple of xi and xj as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (xi,xj) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p2qr, or p3qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying xi< 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.