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An upper bound on the mean value of the Erdős-Hooley Delta function
Published 14 Jun 2023 in math.NT | (2306.08615v4)
Abstract: The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} #{d|n : eu<d\le e{u+1}}$. We prove that $\sum_{n\le x} \Delta(n) \ll x(\log\log x){11/4}$ for all $x\ge100$. This improves on earlier work of Hooley, Hall--Tenenbaum and La Bret`eche-Tenenbaum.
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