On the sum of reciprocals of least common multiples, II

Abstract

Let n and r be positive integers with r≥2 and let A={ai}i=1∞ be a strictly increasing sequence of positive integers. Let SA,r(n):=∑i=1n1lcm(ai,…,ai+r−1). In 1978, Borwein showed that SA,2(n)≤1−12n with the equality occurring if and only if ai=2i−1 for 1≤i≤n+1. In 2017, Qian proved that SA,r(n)≤Vr(n) for 3≤r≤7 and characterized the first n+r−1 terms of the sequence A such that SA,r(n)=Vr(n) holds, where Vr(n) depends only on r and n. In this paper, we further investigate SA,r(n) for 8≤r≤11 and we obtain the least upper bound Ur of SA,r(n) for all strictly increasing sequences A of positive integers and for all positive integers n, where Ur is a constant depending only on r.