Étude statistique du facteur premier médian, 2 : lois locales
Jonathan Rotgé
TL;DR
The paper addresses the local distribution of the middle prime factor of integers, treating separately the multiplicity-inclusive case $\Omega(n)$ and the plain-prime case $\omega(n)$. It develops a refined analytic framework that yields uniform, effective asymptotics for the local counts $M_ω(x,p)$ and $M_Ω(x,p)$ across a wide domain, and reveals a phase transition in the $Ω$-case near the critical parameter $β_p=1/5$. The approach combines generating-function techniques, precise control of auxiliary sums such as $\lambda_Ω(k,y)$, and a careful zone-by-zone analysis (outside and inside the critical window) to obtain sharp error terms and to describe the dominant contributions. The results extend and sharpen prior work on local laws for prime factors, capture the impact of small primes on the median factor, and provide a framework adaptable to $p$-positioned variants like $p_{Ω}^{(α)}(n)$, with potential implications for probabilistic models of prime factorization. Overall, the paper advances the quantitative understanding of how the median prime factor behaves locally, with explicit asymptotics and phase-transition phenomena that are relevant for probabilistic number theory and related applications.
Abstract
We estimate the local laws of the distribution of the middle prime factor of an integer, defined according to multiplicity or not. An asymptotic estimate with effective remainder is provided for a wide range of values. In particular this enables to precisely describe the phase transition occurring in the relevant distribution.