PreprintCC BY 4.0arXiv
On the denominators of signed harmonic sums
Abstract
Let eps=(epsi)i≥1 be a sequence with epsi∈{−1,1}, and let vn(eps) be the positive denominator of ∑i=1nepsi/i in lowest terms. We prove that |{1≤ n≤ x:vn(eps)≠ vn+1(eps)}|=o(x) uniformly in eps. In particular, the set of positive integers n with vn(eps)=vn+1(eps) has asymptotic density one for every sequence of signs. This answers a problem of Wu and Chen, \emph{J. Number Theory} \textbf{200} (2019), 397--406.
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BibTeX
@misc{yang2026denominators,
title = {{On the denominators of signed harmonic sums}},
author = {Quan-Hui Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26622},
eprint = {2610.26622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26622},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the denominators of signed harmonic sums
AU - Quan-Hui Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26622
UR - https://arxiv.org/abs/2610.26622
AB - Let eps=(eps_i)_i≥1 be a sequence with eps_i∈{−1,1}, and let v_n(eps) be the positive denominator of ∑_i=1^neps_i/i in lowest terms. We prove that |{1≤ n≤ x:v_n(eps)≠ v_n+1(eps)}|=o(x) uniformly in eps. In particular, the set of positive integers n with v_n(eps)=v_n+1(eps) has asymptotic density one for every sequence of signs. This answers a problem of Wu and Chen, \emph{J. Number Theory} \textbf{200} (2019), 397--406.
ER -