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arXiv · 2610.28328

A Joint Local Limit Theorem for Prime Divisor Counting Functions

Jovan Andreevski
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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.

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BibTeXRIS

Jovan Andreevski. 2026-10-20. A Joint Local Limit Theorem for Prime Divisor Counting Functions. https://arxiv.org/abs/2610.28328

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