Title: The mean value of the Erd\H os--Hooley Delta function
Abstract: The Erd\H os--Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$. In a seminal 1979 paper, Hooley proved that estimates of its mean value can be exploited to count solutions to certain Diophantine equations. In this talk, I will present recent work, joint with Kevin Ford and Terence Tao, that establishes improved upper and lower bounds on the mean value of Delta.
