Square-root logarithmic bounds for the arithmetic complexity of factorial multiples
Abstract
Assume that the Riemann zeta function has no zeros in ℜ(s)>1−δ, s≠1, for some fixed 0<δ≤1/2. We prove that, for every sufficiently large integer k, a positive multiple mk k! has arithmetic complexity τ(mk k!)≤exp(C√(log kloglog k)), where C depends only on δ. We first establish a smooth-number asymptotic at every sufficiently large center x in intervals of length at least x3/4. Combining this estimate with an established genus-two construction yields the arithmetic bound without Cheng's conjecture on smooth numbers in elliptic Hasse intervals. The bound includes the cost of constructing all integer parameters. This work was carried out with assistance from the UniMath model, developed by UniDT (Shanghai) Co., Ltd., and OpenAI's GPT-6.1 Sol and GPT-6 Astra.
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BibTeX
@misc{zhang2026squareroot,
title = {{Square-root logarithmic bounds for the arithmetic complexity of factorial multiples}},
author = {Liang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36588},
eprint = {2610.36588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36588},
}
RIS
TY - UNPB
TI - Square-root logarithmic bounds for the arithmetic complexity of factorial multiples
AU - Liang Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36588
UR - https://arxiv.org/abs/2610.36588
AB - Assume that the Riemann zeta function has no zeros in ℜ(s)>1−δ, s≠1, for some fixed 0<δ≤1/2. We prove that, for every sufficiently large integer k, a positive multiple m_k k! has arithmetic complexity τ(m_k k!)≤exp(C√(log kloglog k)), where C depends only on δ. We first establish a smooth-number asymptotic at every sufficiently large center x in intervals of length at least x^3/4. Combining this estimate with an established genus-two construction yields the arithmetic bound without Cheng's conjecture on smooth numbers in elliptic Hasse intervals. The bound includes the cost of constructing all integer parameters. This work was carried out with assistance from the UniMath model, developed by UniDT (Shanghai) Co., Ltd., and OpenAI's GPT-6.1 Sol and GPT-6 Astra.
ER -