A lower bound on the mean value of the Erdős-Hooley Delta function

We give an improved lower bound for the average of the Erdős-Hooley function $\mathrm{\Delta }(n)$, namely ${\sum }_{n\le x}\mathrm{\Delta }(n){\gg }_{\epsilon }x(\log \log x{)}^{1+\eta -\epsilon }$ for all $x⩾100$ and any fixed $\epsilon$, 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 $x(\log \log x{)}^{11\mathrm{/}4}$ of the second and third authors.