A Limit Involving Least Common Multiples: 10797

The Least Common Multiple of Consecutive Terms in a Quadratic Progression

“…Hanson [7] and Nair [14] respectively obtained the upper bound and lower bound of lcm 1≤i≤n {i}. Bateman et al [1] obtained an asymptotic estimate for the least common multiple of arithmetic progressions. Recently, Hong et al [10] obtained an asymptotic estimate for the least common multiple of a sequence of products of linear polynomials.…”

The least common multiple of random sets of positive integers

“…Some progress has been made in this direction. While the Prime Number Theorem is equivalent to the asymptotic ψ q (n) ∼ n for q(x) = x, Paul Bateman noticed that the Prime Number Theorem for arithmetic progressions could be exploited to obtain the asymptotic estimate when q(x) = a 1 x + a 0 is a linear polynomial and proposed it as a problem [1] in the American Mathematical Monthly:…”

New lower bounds for the least common multiples of arithmetic progressions

“…In [4], Farhi provided an identity involving the least common multiple of binomial coefficients and then use it to give a simple proof of the estimate (1.1). Inspired by Hanson's and Nair's works, Bateman, Kalb, and Stenger [1] and Farhi [2] respectively sought asymptotics and nontrivial lower bounds for the least common multiples of arithmetic progressions. Recently, Hong, Qian and Tan [10] extended the Bateman-Kalb-Stenger theorem from the linear polynomial to the product of linear polynomials.…”