An upper bound on the mean value of the Erd\H{o}s-Hooley Delta function

Dimitris Koukoulopoulos; Terence Tao

doi:10.1112/plms.12572

An upper bound on the mean value of the Erd\H{o}s-Hooley Delta function

Number Theory 2023-12-11 v4

Authors: Dimitris Koukoulopoulos , Terence Tao

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Abstract

The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$ . 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.

Keywords

erős-hooley delta function mean value upper bound divisor function analytic number theory

Cite

@article{arxiv.2306.08615,
  title  = {An upper bound on the mean value of the Erd\H{o}s-Hooley Delta function},
  author = {Dimitris Koukoulopoulos and Terence Tao},
  journal= {arXiv preprint arXiv:2306.08615},
  year   = {2023}
}

Comments

18 pages. Some minor changes. This is the final version that will apper in Proc. London Math. Soc

An upper bound on the mean value of the Erd\H{o}s-Hooley Delta function

Abstract

Keywords

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