On the representation of friable integers by linear forms

Armand Lachand¹

¹ Institut Élie Cartan Université de Lorraine B.P. 70239 54506 Vandœuvre-lès-Nancy Cedex, France

Acta Arithmetica, Tome 181 (2017) no. 2, pp. 97-109.

Résumé

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.

Détail

DOI : 10.4064/aa8153-9-2017

Keywords: denote largest prime factor integer using nilpotent hardy littlewood method developed green tao asymptotic formula varpsi cdots mathcal cap n boldsymbol mathcal cap n vphantom boldsymbol cdots boldsymbol leq boldsymbol cdots boldsymbol leq where ldots system affine linear forms mathbb ldots which affinely related mathcal convex body improves balog blomer dartyge tenenbaum work nbsp comment math helv products linear forms