Asymptotic Behavior of Eigenvalues of Greatest Common Divisor Matrices

Improvements of lower bounds for the least common multiple of finite arithmetic progressions

“…Recently, Green and Tao [5] proved a significant theorem saying that the set of primes contains arbitrarily long arithmetic progressions. On the other hand, Bachman and Kessler [2] and Myerson and Sander [11] investigated the divisibility properties of lcm{1, · · · , n} (i.e., the least common multiple of all elements in the set {1, ..., n}) while Hong and Loewy [9] studied asymptotic behavior of eigenvalues of Smith matrices defined on arithmetic progressions.…”

Asymptotic Behavior of Eigenvalues of Reciprocal Power LCM Matrices

“…From this, one can only conclude that its eigenvalues are positive. Hong and Loewy [24,25] investigated the eigen-structure of the power GCD matrices and made some significant progress in this topic. In fact, it was proved in [24] that if 0 < r ≤ 1 and q ≥ 1 is any fixed integer, then the q-th smallest eigenvalue of the n × n power GCD matrix ((i, j) r ) approaches zero as n tends to infinity.…”

“…Hong and Loewy [24,25] investigated the eigen-structure of the power GCD matrices and made some significant progress in this topic. In fact, it was proved in [24] that if 0 < r ≤ 1 and q ≥ 1 is any fixed integer, then the q-th smallest eigenvalue of the n × n power GCD matrix ((i, j) r ) approaches zero as n tends to infinity. We also note that Cao [8], Hong [20], Hong-Shum-Sun [26] and Li [30] investigated the nonsingularity of power LCM matrices while Hong [18,19], Haukkanen-Korkee [11] and Zhao-Hong-Liao-Shum [37] studied the divisibility of power LCM matrices by power GCD matrices.…”

“…Associated to an arbitrary given strictly increasing infinite sequence of positive integers, we can form an infinite sequence of power GCD matrices and an infinite sequence of reciprocal power LCM matrices. Hong and Loewy [24] proved that the sequence of the smallest eigenvalues of such power GCD matrices converges and the limit is nonnegative when n approaches infinity. But the limit may be positive (see Theorem 2.5 of [24]).…”

“…Hong and Loewy [24] proved that the sequence of the smallest eigenvalues of such power GCD matrices converges and the limit is nonnegative when n approaches infinity. But the limit may be positive (see Theorem 2.5 of [24]). However, by Theorem 2.2 (ii) we know that the limit of the smallest eigenvalue of such reciprocal power LCM matrices is zero as n goes to infinity.…”