Problem 317

Erdős Problem #317: declared status 'open'. Formalized: yes. 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? Current best: 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. Prize: no. Tags: number theory, unit fractions.