Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables

Abstract

For $Q$ a polynomial with integer coefficients and $x, y \geq 2$, we prove upper bounds for the quantity $\Psi_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leq y\}|$. We apply our results to a problem of De Koninck, Doyon and Luca on integers divisible by the square of their largest prime factor. As a corollary to our arguments, we improve the known level of distribution of the set $\{n^2-D\}$ for well-factorable moduli, previously due to Iwaniec. We also consider the Chebyshev problem of estimating $\max\{P^+(n^2-D), n\leq x\}$ and make explicit, in Deshouillers-Iwaniec's state-of-the-art result, the dependence on the Selberg eigenvalue conjecture.