PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Abstract

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.

Authors

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image