# Niveau de r\'epartition des polyn\^omes quadratiques et crible majorant pour les entiers friables

## TL;DR

This paper improves estimates on the distribution of quadratic polynomial values for well-factorable moduli, explores related Chebyshev problems, and develops new bounds for numbers with restricted prime factors in polynomial sequences.

## Contribution

It introduces new bounds on the level of distribution for quadratic polynomial values and explicit dependence on the Selberg eigenvalue conjecture, advancing sieve methods for polynomial sequences.

## Key findings

- Enhanced distribution estimates for quadratic polynomial values.
- Explicit bounds involving the Selberg eigenvalue conjecture.
- New upper bounds for numbers with small prime factors in polynomial sequences.

## 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 $\max\{P^+(n^2-D), n\leq x\}$ 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 $\Psi_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leq y\}|$ for $Q\in{\mathbb Z}[X]$ linear or quadratic.