PreprintCC BY 4.0arXiv
A Joint Local Limit Theorem for Prime Divisor Counting Functions
Abstract
We establish a uniform two-dimensional local limit theorem for the joint distribution of Ω(n) and Ω(n+1). More precisely, along a set of scales of logarithmic density one, the normalized joint counts are uniformly approximated by the density of two independent Gaussian random variables, each with mean and variance loglog x, up to a power-saving error. The proof builds upon a recent circle-method framework of Tao and Teräväinen. Our results equally hold for the additive function ω. Additionally, we explore applications of our local limit theorem in the setting of uniquely ergodic systems.
Related
- Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
- Shifted convolutions of divisor functions
- Statistics of divisor functions over function fields
- Triple divisor type functions in arithmetic progressions to prime power moduli
- Additive Sums of Generalized Shifted Ternary Divisor Functions
BibTeX
@misc{andreevski2026joint,
title = {{A Joint Local Limit Theorem for Prime Divisor Counting Functions}},
author = {Jovan Andreevski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28328},
eprint = {2610.28328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28328},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Joint Local Limit Theorem for Prime Divisor Counting Functions
AU - Jovan Andreevski
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28328
UR - https://arxiv.org/abs/2610.28328
AB - We establish a uniform two-dimensional local limit theorem for the joint distribution of Ω(n) and Ω(n+1). More precisely, along a set of scales of logarithmic density one, the normalized joint counts are uniformly approximated by the density of two independent Gaussian random variables, each with mean and variance loglog x, up to a power-saving error. The proof builds upon a recent circle-method framework of Tao and Teräväinen. Our results equally hold for the additive function ω. Additionally, we explore applications of our local limit theorem in the setting of uniquely ergodic systems.
ER -