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

TL;DR

This paper establishes an improved lower bound for the average of the Erd ext{o}s-Hooley delta function, refining previous bounds and contributing to the understanding of its asymptotic behavior.

Contribution

It provides a sharper lower bound for the mean value of the Erd ext{o}s-Hooley delta function, advancing prior results and connecting to recent upper bounds.

Findings

Lower bound: x(\u2212 0aa)

Improved over previous x \u2212aaaaaaaa

Related to recent upper bounds of x(aaaa)^{11/4}

Abstract

We give an improved lower bound for the average of the Erd\H{o}s-Hooley function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and third authors.