Two upper bounds for the Erd\H{o}s--Hooley Delta-function

TL;DR

This paper improves the upper bounds on the average and typical size of the Erd ext{"o}s--Hooley Delta-function, which measures the distribution of divisors of integers within exponential intervals.

Contribution

It provides new, tighter upper bounds for the average and normal orders of the Erd ext{"o}s--Hooley Delta-function, advancing understanding of its behavior.

Findings

Improved upper bounds for the average order of elta(n).

Enhanced bounds for the normal order of elta(n).

Refined estimates contribute to divisor distribution analysis.

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 improve the current upper bounds for the average and normal orders of this arithmetic function.