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

Abstract

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

15 pages. Added remarks in the end of page 3; added references [3] and [9]. Final version, to appear in Proc. London Math. Soc