arXiv subjects

Alexandre Lartaux

Publications and source records attributed to Alexandre Lartaux.

3 recordsLinked to original sources

arXiv2021

Sur la fonction Delta de Hooley associée à des caractères

Let $(f_1,f_2)$ a $2$-tuple of arithmetic functions and $$Δ_3(n,f_1,f_2):=\sup\limits_{\substack{(u_1,u_2) \in \mathbb{R}^{2} \\(v_1,v_2) \in [0,1]^{2}}}\Big\lvert \sum\limits_{\substack{d_1 d_{2} \mid n \\ e^{u_i}<d_i\leqslant e^{u_i+v_i}}}{f_1(d_1) f_{2}(d_{2})} \Big\rvert{\rm .}$$ In this paper, we give an upper bound of the second moment of $Δ_3(n,χ_1,χ_2)$ when $χ_1$ and $χ_2$ are two non principal Dirichlet characters, following methods developed by La Bretèche and Tenenbaum. This upper bound is a main step for the asymptotic counting of the number of ideals of norm fixed, which will be developped in another article.

math.NT↗

arXiv2021

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

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

math.NT↗

arXiv2020

Moments de la fonction Delta de Hooley associée à un caractère

Let $f$ be an arithmetic function, $V\geqslant 1$ a real number and $$Δ_V(n,f) := \sup\limits_{\substack{u \in \mathbb{R}\\ v \in [0,V]}}{\Big|\sum\limits_{\substack{d\mid n \\ {\rm e}^{u}<d\leqslant {\rm e}^{u+v}}}{f(d)}\Big|}{\rm .}$$ In a paper published in 2012, La Bretèche and Tenenbaum investigated weighted moments of $Δ_1(n,f)$ where $f$ is a non principal real Dirichlet character, or the Möbius function. Answering a question of Hooley, we extend their results studying dependance in $V$ and including the case of complex characters.

math.NT↗