Étude statistique du facteur premier médian, 3 : lois de répartition
Jonathan Rotgé
TL;DR
The paper establishes a uniform Gaussian limit for the distribution of the middle prime factor $p_{m,\\nu}(n)$, with $\\nu\\in\\{\\omega,\\Omega\\}$, by proving $\\mathcal{A}_\\nu(x,t)=\\Phi(2t)+O(1/\\sqrt{\\log_2 x})$ as $x\\to\\infty$ for all real $t$. It combines sharp local laws for the local distribution of $p_{m,\\nu}(n)$ with a careful decomposition over prime ranges, converting the main term into a Stieltjes integral and controlling error terms via Turán-Kubilius-type bounds and Mertens-type estimates. The work refines previous Gaussian-limit results and provides an optimal error rate, improving the understanding of the normal behavior of the median prime factor. The methods have potential implications for finer probabilistic models of integer factorization and related arithmetic functions.
Abstract
We consider the Gaussian limit law for the distribution of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an optimal bound for the speed of convergence, thereby improving on previous estimates available in the literature.