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arXiv · 1806.05402

Small values of signed harmonic sums

Sandro Bettin·Giuseppe Molteni·Carlo Sanna
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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 $s_1, \ldots, s_n \in \{-1, +1\}$. We prove that $\mathfrak{m}_N(\tau) < \exp\!\big(-C(\log N)^2\big)$, for all sufficiently large positive integers $N$ (depending on $C$ and $\tau$), where $C$ is any positive constant less than $1/\log 4$.

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Sandro Bettin, Giuseppe Molteni, Carlo Sanna. 2018-06-14. Small values of signed harmonic sums. https://arxiv.org/abs/1806.05402

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