\'Etude statistique du facteur premier m\'edian, 2 : lois locales

TL;DR

This paper investigates the local distribution laws of the middle prime factor of integers, providing asymptotic estimates and describing phase transitions in the distribution.

Contribution

It offers new asymptotic estimates with effective remainders for the local laws of the middle prime factor distribution, including phase transition analysis.

Findings

Asymptotic estimates with effective remainders for the distribution

Precise description of phase transition in the distribution

Analysis applicable to various multiplicity definitions

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.