OPEN
For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with
\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]
Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?
Erdős
[Er50c] proved that
\[\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.\]
The upper bound was improved by Vose
[Vo85] to
\[N(b) \ll \sqrt{\log b}.\]
One can also investigate the average of $N(a,b)$ for fixed $b$, and it is known that
\[\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.\]
Related to [18].
