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

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 ${\sum }_{\begin{array}{c}\mathrm{\mid }{n}_1\mathrm{\mid }\le x,\mathrm{\mid }{n}_2\mathrm{\mid }\le x\end{array}}h(F(n_1,n_2))$ 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 $\exp \left(\frac{\log x}{(\log \log x{)}^{1\mathrm{/}2-\epsilon }}\right)\le y\le x$. This improves a result of Balog, Blomer, Dartyge and Tenenbaum (2012).