Small values of signed harmonic sums

Abstract

For every $\tau \in \mathbb{R}$ and every integer $N$, let ${\mathfrak{m}}_N(\tau )$ be the minimum of the distance of $\tau$ from the sums ${\sum }_{n=1}^N{s}_n/n$, where $s_1,\dots ,s_n\in \{-1,+1\}$. We prove that ${\mathfrak{m}}_N(\tau )<\exp (-C(\log N{)}^2)$, for all sufficiently large positive integers $N$ (depending on $C$ and $\tau$), where $C$ is any positive constant less than $1/\log 4$.

minor corrections to the text, a small improvement to Proposition 2.7