The sum of reciprocals of least common multiples

Abstract

For a strictly increasing sequence $A=\{a_i\}_{i=1}^{\infty }$ of positive integers, Borwein (Can Math Bull 20:117–118, 1978) proved that $\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i,a_{i+1})}^{-1}\le 1-\frac{1}{2^n}$ for any positive integer n, where the equality holds if and only if $a_i=2^{i-1}$ for $i=1,2, \ldots ,n+1$. Let r be an integer with $3\le r\le 7$, Qian (C R Acad Sci Paris Ser I 355:1127–1132, 2017) further proved that $\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1})}^{-1}\le U_r(n)$ and characterized the equality, where $U_r(n)$ depends only on r and n. In this paper, under the condition ${{\,\mathrm{lcm}\,}}(a_1, \ldots ,a_{r-1})\le a_r$, we determine the best upper bound (uniformly dependent only on r and n) of $\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1})}^{-1}$ and also characterize the terms $a_1,a_2, \ldots ,a_{n+r-1}$ such that the best upper bound is attained.