Year: 2026
Type: preprint
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 $\log\log x$, 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.
Source: arXiv (Cornell University)
Author: Andreevski, Jovan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223139907