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

Let $\alpha \in ]0,1]$ and let $Q_j$ $(1⩽j⩽r)$ 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 $x<n⩽x+x^{\alpha }$ 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 $Q_j$.