On moments of the Erd\H{o}s--Hooley Delta-function

TL;DR

This paper investigates the Erdős–Hooley Delta-function, providing new upper bounds for its weighted real moments, which enhances understanding of its distribution and extremal behavior.

Contribution

The paper introduces novel upper bounds for weighted real moments of the Erdős–Hooley Delta-function, advancing previous estimates and analytical techniques.

Findings

Established tighter upper bounds for moments of the Delta-function

Improved understanding of the distribution of divisors in short intervals

Enhanced analytical methods for studying divisor-related functions

Abstract

For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We provide new upper bounds for weighted real moments of this function.