\'Etude statistique du facteur premier m\'edian, 1 : valeur moyenne

TL;DR

This paper derives an improved asymptotic expansion for the average logarithm of the median prime factor of integers, enhancing previous estimates with an optimal remainder term.

Contribution

It generalizes recent work by providing a more precise asymptotic expansion for the mean value of the median prime factor's logarithm.

Findings

Enhanced asymptotic estimate with optimal remainder term

Generalization of previous median prime factor studies

Improved understanding of prime factor distribution

Abstract

We provide an asymptotic expansion for the mean-value of the logarithm of the middle prime factor of an integer, defined according to multiplicity or not, thus generalising a recent study of McNew, Pollack, and Singha Roy. This yields an improvement of the asymptotic estimate, in particular by furnishing an optimal remainder when the expansion is truncated at the first order.