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

TL;DR

This paper studies the average behavior of arithmetic functions evaluated at values of an irreducible binary cubic form and provides asymptotic estimates for the distribution of y-friable values, improving previous results.

Contribution

It offers new estimates for the average order of functions over binary cubic form values and asymptotic formulas for y-friable values, extending prior work in the field.

Findings

Derived estimates for sums involving binary cubic forms.

Provided asymptotic formulas for y-friable values of the form.

Improved bounds compared to previous research.

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).