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On the representation of friable integers by linear forms

Volume 181 / 2017

Armand Lachand Acta Arithmetica 181 (2017), 97-109 MSC: Primary 11N25; Secondary 11N37. DOI: 10.4064/aa8153-9-2017 Published online: 6 November 2017


Let $P^+(n)$ denote the largest prime factor of the integer $n$. Using the nilpotent Hardy–Littlewood method developed by Green and Tao, we give an asymptotic formula for $$ \varPsi_{F_1\cdots F_t}(\mathcal{K}\cap[-N,N]^d,N^{1/u}) := \#\{\boldsymbol{n}\in \mathcal{K}\cap[-N,N]^d:\vphantom{P^+(F_1(\boldsymbol{n})\cdots F_t(\boldsymbol{n}))\leq N^{1/u}} P^+(F_1(\boldsymbol{n})\cdots F_t(\boldsymbol{n}))\leq N^{1/u}\} $$ where $(F_1,\ldots,F_t)$ is a system of affine-linear forms on $\mathbb{Z}[X_1,\ldots,X_d]$ no two of which are affinely related, and $\mathcal{K}$ is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum’s work [Comment. Math. Helv. 87 (2012)] in the case of products of linear forms.


  • Armand LachandInstitut Élie Cartan
    Université de Lorraine
    B.P. 70239
    54506 Vandœuvre-lès-Nancy Cedex, France

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