Sur le nombre d’idéaux dont la norme est la valeur d’une forme binaire de degré 3

Abstract Let $\mathbb{K}$ be a cyclic extension of degree 3 $\mathbb{Q}$. Take $G={\rm Gal}(\mathbb{K}/ \mathbb{Q})$ and χ the character non trivial representation G. In this case, is non-principal Dirichlet quantity $r_3(n)$ defined by $$r_3(n):=\big(1*\chi*\chi^2\big)(n)$$ counts number ideals $O_{\mathbb{K}}$ norm n. paper, using new result on Hooley’s Delta function from [11], we prove an asymptotic estimate, in ξ, $$Q(\xi,\mathcal{R},F):=\sum\limits_{\boldsymbol{x} \in \mathcal{R}(\xi)}{r_3\big(F(\boldsymbol{x})\big)}{\rm ,}$$ for binary form F irreducible over $\mathcal{R}$ good domain $\mathbb{R}^2$, with $$\mathcal{R}(\xi):=\Big\{\boldsymbol{x} \mathbb{R}^2\;:\: \frac{\boldsymbol{x}}{\xi} \mathcal{R}\Big\}{\rm .}$$ We also give geometric interpretation main constant estimate when ring principal.