Egyptian fractions of bounded length

期刊论文
发表日期：

2024年1月22日

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Noah Lebowitz-Lockard

摘要

Let $A_k (n)$ $A_{k} (n)$ be the number of $a < n$ $a < n$ coprime to n for which the equation $a/n = 1/m_1 + \cdots + 1/m_k$ $a / n = 1 / m_{1} + \dots + 1 / m_{k}$ has a solution. We show that for a fixed $k \ge 3$ $k \geq 3$ , the sum of $A_k (n)$ $A_{k} (n)$ over all $n \le x$ $n \leq x$ is $\gg x(\log x)^{2k - 1}$ $≫ x {(log x)}^{2 k - 1}$ . In the $k = 3$ $k = 3$ case, we find a new upper bound for this sum as well as a new upper bound for individual values of $A_3 (n)$ $A_{3} (n)$ .