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The shortest harmonic sums with decreasing denominator
Abstract
For a positive integer a, let b(a) be the smallest integer b > a such that the denominator of 1/a + 1/(a+1) + ⋯ + 1/b is smaller than the denominator of 1/a + 1/(a+1) + ⋯ + 1/(b−1). Recently it was shown that the limit inferior liminfa → ∞ ((b(a) − a)/(log a)) exists and is positive. Here we find its exact value.
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BibTeX
@misc{doorn2026shortest,
title = {{The shortest harmonic sums with decreasing denominator}},
author = {Wouter van Doorn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00104},
eprint = {2609.00104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00104},
}
RIS
TY - UNPB
TI - The shortest harmonic sums with decreasing denominator
AU - Wouter van Doorn
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00104
UR - https://arxiv.org/abs/2609.00104
AB - For a positive integer a, let b(a) be the smallest integer b > a such that the denominator of 1/a + 1/(a+1) + ⋯ + 1/b is smaller than the denominator of 1/a + 1/(a+1) + ⋯ + 1/(b−1). Recently it was shown that the limit inferior liminf_a → ∞ ((b(a) − a)/(log a)) exists and is positive. Here we find its exact value.
ER -