Moments de la fonction Delta de Hooley associ\'ee \`a un caract\`ere

Abstract

Let $f$ be an arithmetic function, $V\geqslant 1$ a real number and $$\Delta_V(n,f) := \sup\limits_{\substack{u \in \mathbb{R}\\ v \in [0,V]}}{\Big|\sum\limits_{\substack{d\mid n \\ {\rm e}^{u}<d\leqslant {\rm e}^{u+v}}}{f(d)}\Big|}{\rm .}$$ In a paper published in 2012, La Bret\`eche and Tenenbaum investigated weighted moments of $\Delta_1(n,f)$ where $f$ is a non principal real Dirichlet character, or the M\"obius function. Answering a question of Hooley, we extend their results studying dependance in $V$ and including the case of complex characters.