A lower bound on the mean value of the Erdős-Hooley delta function (Q6581868)

The Erdős-Hooley Delta function is defined for a natural number \(n\) as\N\[\N\Delta(n):=\max_{u \in \mathbb{R}} \#\left\{d|n: e^u<d\le e^{u+1}\right\}.\N\]\NLetting \(\mathrm{Log}\,x:=\max\{1, \log x\}\) for \(x>0\), and also \(\mathrm{Log}_2 x :=\mathrm{Log}(\mathrm{Log}\,x)\), in the paper under review, the authors prove that for any \(\varepsilon>0\) and all \(x\ge 1\), one has\N\[\N\sum_{n\le x}\Delta(n) \gg_\varepsilon x (\mathrm{Log}_2 x)^{1+\eta-\varepsilon},\N\]\Nwhere \(\eta=\frac{\log 2}{\log(2/\rho)}=0.353327\dots\) and \(\rho\) is the unique number in \((0,1/3)\) satisfying the equation \(1 - \rho/2 = \lim_{j\to\infty} 2^{j-2}/\log a_j\) with \(a_1 = 2\), \(a_2 = 2 + 2^{\rho}\) and \(a_j = a_{j-1}^2 + a_{j-1}^{\rho} - a_{j-2}^{2\rho}\) for \(j\in\mathbb{Z}_{\ge3}\). This improves on the lower bound \(x\mathrm{Log}_2 x\) obtained by \textit{R. R. Hall} and \textit{G. Tenenbaum} [J. Lond. Math. Soc., II. Ser. 25, 392--406 (1982; Zbl 0489.10046)], and can be compared to the recent upper bound of \(x (\mathrm{Log}_2 x)^{11/4}\) obtained by the second and third authors of the present paper in [Proc. Lond. Math. Soc. (3) 127, No. 6, 1865--1885 (2023; Zbl 1536.11142)].