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Current version
For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that\[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erdős
[BlEr76] have shown that\[D(b)\ll b(\log b)^2.\]If $b=p$ is a prime then\[D(p) \gg p\log p.\]This was solved by Yokota
[Yo88], who proved that\[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\]This was improved by Liu and Sawhney
[LiSa24] to\[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
2025-10-20 00:00:00
For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that\[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erdős
[BlEr76] have shown that\[D(b)\ll b(\log b)^2.\]If $b=p$ is a prime then\[D(p) \gg p\log p.\]
This was solved by Yokota
[Yo88], who proved that\[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\]This was improved by Liu and Sawhney
[LiSa24] to\[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
