On the representation of friable integers by linear forms

Armand Lachand (1)

1 Analyse

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the so-called nilpotent Hardy-Littlewood method developped by Green and Tao, we give an asymptotic formula for the cardinal $\Psi_{F_1\cdots F_t}\left(\mathcal{K}\cap [−N, N ]^d , N^{ 1/u}\right) := \# \left\{n ∈ \mathcal{K}\cap [−N, N ]^d : P^+(F_1(n)\cdots F_t(n)) \leqslant N^{1/u}\right\}$ where $(F_1,\cdots , F_t)$ is a system of affine-linear forms of $\mathbf{Z}[X_1 ,\cdots , X_d ]$ no two of which are affinely related and $\mathcal{K}$ is a convex body. This improves a work of Balog, Blomer, Dartyge and Tenenbaum [BBDT12] in the case of product of linear forms.

Domaines: Théorie des nombres [math.NT]

Armand Lachand. On the representation of friable integers by linear forms. 2014. ⟨hal-01081277v1⟩