Problem Statement
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with\[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$,\[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\]whenever the left-hand side is not zero?
Categories:
Number Theory Unit Fractions
Progress
Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example\[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\]Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of\[2^{-n\frac{(\log\log\log n)^{1+o(1)}}{\log n}},\]and van Doorn gives a heuristic that suggests this may be the true order of magnitude.Source: erdosproblems.com/317 | Last verified: January 14, 2026