Note sur les lois locales conjointes de la fonction nombre de facteurs premiers

TL;DR

This paper investigates the local joint distribution of the number of prime factors of polynomial values within short intervals, showing it is dominated by an independent model with explicit bounds.

Contribution

It establishes a majorization result for the joint distribution of prime factors of polynomial values in short intervals, with explicit bounds depending on polynomial coefficients.

Findings

Joint distribution is majorized by an independent model.

Provides explicit bounds for the majorization constant.

Results apply to polynomials with integer coefficients in short intervals.

Abstract

Let $\alpha\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^\alpha$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.