Fonctions arithm\'etiques et formes binaires irr\'eductibles de degr\'e $3$

Abstract

Let $F(X_1,X_2)\in\mathbb{Z}[X_1,X_2] $ be an irreducible binary form of degree $3$ and $h$ an arithmetic function. We give some estimates for the average order $\sum_{\substack{|n_1|\leq x,|n_2|\leq x}}h(F(n_1,n_2))$ when $h$ satisfy certain conditions. As an application, we provide some asymptotic formula for the number of $y$-friable values of $F(n_1,n_2)$ when the variables $n_1,n_2$ lies in the square $[1,x]^2$ and uniformly in the region $\exp\left(\frac{\log x}{(\log\log x)^{1/2-\varepsilon}}\right)\leq y\leq x$. This improves a result of Balog, Blomer, Dartyge and Tenenbaum (2012).