A sharp upper bound for the sum of reciprocals of least common multiples

Abstract

Let $n$ and $k$ be positive integers such that $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be an arbitrary given strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} \frac{1}{ 1{\rm cm} (a_{i},a_{i+k})}$. Borwein [3] proved a conjecture of Erdős stating that if $n\ge 2$, then $S_{n,1}\le 1-\frac{1}{2^{n-1}}$, with the equality holding if and only if $a_{i}=2^{i-1}$ for $1\le i \le n$. In this paper, we first improve Borwein's upper bound by showing that $S_{n,1}\le \frac{1}{a_{1}}(1-\frac{1}{2^{n-1}})$ with the equality occurring if and only if $a_{i}=2^{i-1}a_{1}$ for all integers $1 \le i \le n$. Then we use this improved upper bound to show that if $n\ge 3$, then $S_{n, 2}\le \frac{7}{6}+\frac{1}{2^{\lfloor \frac{n}{2}\rfloor }} (\frac{2}{3}\delta _{n}-\frac{7}{3})$, with the equality holding if and only if $a_1=1, a_{2i}=2^i$ and $a_{2i+1}=3\times 2^{i-1}$ for all integers $1\le i\le \lfloor \frac{n}{2}\rfloor$, where $\delta _{n}:=0$ if $n$ is even, and 1 if $n$ is odd. Furthermore, we show that if $n\ge 7$, then $S_{n, 3}\le \frac{17}{15}-\frac{37}{15}\cdot \frac{1}{2^{\lfloor \frac{n}{3}\rfloor }} +\frac{\epsilon _{n}}{2^{\lceil \frac{n}{3}\rceil }}$, with equality occurring if and only if $a_i=i$ for all $i\in \{1, 2, 3\}$ and $a_{3i+1}=2^{i+1} (1\le i\le \lfloor \frac{n-1}{3}\rfloor ), a_{3i+2}= 5\times 2^{i-1} (1\le i\le \lfloor \frac{n-2}{3}\rfloor )$ and $a_{3i+3}=3\times 2^i (1\le i\le \lfloor \frac{n}{3}\rfloor -1)$, where $\epsilon _n=0$ if $3 \mid n$, 1 if $n\equiv 1~({\rm mod} \; 3)$ and $\frac{9}{5}$ if $n\equiv 2~({\rm mod} \; 3)$. We also present a tight upper bound for $S_{n, 3}$ if $n\in \{4,5,6\}$.