Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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.