Sur la fonction Delta de Hooley associ\'ee \`a des caract\`eres

Let $(f_1,f_2)$ a $2$-tuple of arithmetic functions and $$\Delta_3(n,f_1,f_2):=\sup\limits_{\substack{(u_1,u_2) \in \mathbb{R}^{2} \\(v_1,v_2) \in [0,1]^{2}}}\Big\lvert \sum\limits_{\substack{d_1 d_{2} \mid n \\ e^{u_i}<d_i\leqslant e^{u_i+v_i}}}{f_1(d_1) f_{2}(d_{2})} \Big\rvert{\rm .}$$ In this paper, we give an upper bound of the second moment of $\Delta_3(n,\chi_1,\chi_2)$ when $\chi_1$ and $\chi_2$ are two non principal Dirichlet characters, following methods developed by La Bret\`eche 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.