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

Abstract

We obtain new estimates on the level of distribution of the set $\{Q(n)\}$ where $Q\in \mathbb{Z}[X]$ is irreducible quadratic, for well-factorable moduli, improving a result due to Iwaniec. As a by-product of our arguments, we study the Chebyshev problem of estimating max$\{P^{+}(n^2-D),n\le x\}$ and make explicit, in Deshouillers–Iwaniec's state-of-the-art result, the dependence on the Selberg eigenvalue conjecture.

Combined with the construction of an upper-bound sieve for numbers free of large factors, we obtain new upper bounds for the quantity ${\Psi }_Q(x,y)=|\{n\le x:p\mid Q(n)\Rightarrow p\le y\}|$ for $Q\in \mathbb{Z}[X]$ linear or quadratic.