An upper bound on the mean value of the Erdős-Hooley Delta function

Abstract

The Erdős-Hooley Delta function is defined for $n\in \mathbb{N}$ as $\Delta (n)={\sup }_{u\in \mathbb{R}}\#\{d|n:e^u<d\le e^{u+1}\}$. We prove that ${\sum }_{n\le x}\Delta (n)\ll x(\log \log x{)}^{11/4}$ for all $x\ge 100$. This improves on earlier work of Hooley, Hall--Tenenbaum and La Bretèche-Tenenbaum.

18 pages. Some minor changes. This is the final version that will apper in Proc. London Math. Soc

References in corpus (1)

• An upper bound on the mean value of the Erdős-Hooley Delta function

Cited by in corpus (2)

• An upper bound on the mean value of the Erdős-Hooley Delta function
• Mean values of arithmetic functions and application to sums of powers