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A Central Limit Theorem for Coprime Pair Counts in Randomly Translated Disks
Abstract
We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to σ2rlog r and establish a central limit theorem under the corresponding normalization, where σ2 is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.
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BibTeX
@misc{ji2026central,
title = {{A Central Limit Theorem for Coprime Pair Counts in Randomly Translated Disks}},
author = {Xin Hang Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14224},
eprint = {2609.14224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14224},
}
RIS
TY - UNPB
TI - A Central Limit Theorem for Coprime Pair Counts in Randomly Translated Disks
AU - Xin Hang Ji
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14224
UR - https://arxiv.org/abs/2609.14224
AB - We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to σ^2rlog r and establish a central limit theorem under the corresponding normalization, where σ^2 is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.
ER -