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 $\Omega(n)$ and $\Omega(n+1)$. 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 Ter\"av\"ainen. Our results equally hold for the additive function $\omega$. Additionally, we explore applications of our local limit theorem in the setting of uniquely ergodic systems.