Note on the mean value of the Erdős--Hooley Delta-function

Régis de la Bretèche, Gérald Tenenbaum

Published 2023-09-07Version 1

For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We improve a recent upper bound by Koukoulopoulos and Tao by showing that $\sum_{n\leqslant x}\Delta(n)\leqslant x(\log_2x)^{2+o(1)}$.