On differences of two harmonic numbers
Creators
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1.
California Institute of Technology
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2.
University of Washington
Abstract
We prove the existence of infinitely many (mk,nk) ∈ N² such that the difference of harmonic numbers Hmk – Hnk approximates 1 well lim k→∞ ∣∑ℓ=nmk1ℓ−1 ∣⋅n²k=0. This answers a question of Erdős 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 min1≤n≤N min m∈N ∣n²θ−m∣.
Copyright and License
© 2025 The Author(s). The publishing rights in this article are licensed to University College London under an exclusive licence. Mathematika is published by the London Mathematical Society on behalf of University College London.
Acknowledgement
JL was partially supported by an NUS Overseas Graduate Scholarship. SS was partially supported by the NSF (DMS-212322) and is grateful for discussions with Vjeko Kovac.
Additional details
Related works
- Is new version of
- Discussion Paper: arXiv:2405.11354 (arXiv)
Funding
- National University of Singapore
- NUS Overseas Graduate Scholarship
- National Science Foundation
- DMS‐212322
Dates
- Accepted
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2024-12-05Accepted
- Available
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2025-01-27Version of Record online