We obtain new estimates on the level of distribution of the set \{Q(n)\} where Q∈ 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≤ 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 Ψ_Q(x, y) = |\{n≤ x: p Q(n)⇒ p≤ y\}| for Q∈ Z[X] linear or quadratic.
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Bretèche et al. (2020) studied this question.