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 (Formula presented.), namely (Formula presented.) for all (Formula presented.) and any fixed (Formula presented.), where (Formula presented.) is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of (Formula presented.) of Hall and Tenenbaum, and can be compared to the recent upper bound of (Formula presented.) of the second and third authors.