Two upper bounds for the Erdős--Hooley Delta-function

Régis de la Bretèche, Gérald Tenenbaum

For integer $n⩾1$ and real $u$, let $\Delta (n,u):=|\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}|$. The Erdős--Hooley Delta-function is then defined by $Δ(n):=\max_{u\in{\…