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arXiv · 2405.11354

On differences of two harmonic numbers

Jeck Lim·Stefan Steinerberger
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Abstract

We prove that the existence of infinitely many $(m_k, n_k) \in \mathbb{N}^2$ such that the difference of harmonic numbers $H_{m_k} - H_{n_k}$ approximates 1 well $$ \lim_{k \rightarrow \infty} \left| \sum_{\ell = n}^{m_k} \frac{1}{\ell} - 1 \right|\cdot n_k^2 = 0.$$ This answers a question of Erd\H{o}s and Graham. The construction uses asymptotics for harmonic numbers, the precise nature of the continued fraction expansion of $e$ and a suitable rescaling of a subsequence of convergents. We also prove a quantitative rate by appealing to techniques of Heilbronn, Danicic, Harman, Hooley and others regarding $\min_{1 \leq n \leq N} \min_{m \in \mathbb{N}}\| n^2 \theta - m\|$.

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Jeck Lim, Stefan Steinerberger. 2024-05-18. On differences of two harmonic numbers. https://arxiv.org/abs/2405.11354

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