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arXiv · 1703.03197

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

Régis de la Bretèche·Sary Drappeau
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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.

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Régis de la Bretèche, Sary Drappeau. 2017-03-09. Niveau de r\'epartition des polyn\^omes quadratiques et crible majorant pour les entiers friables. https://arxiv.org/abs/1703.03197

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