On the friable mean-value of the Erd\H{o}s-Hooley Delta function

TL;DR

This paper establishes uniform bounds for the average behavior of the Erdős-Hooley Delta function over friable integers, enhancing understanding of their distribution and divisor structure.

Contribution

It provides the first uniform upper and lower bounds for the mean-value of the Erdős-Hooley Delta function over friable integers, advancing the analysis of their divisor properties.

Findings

Established uniform bounds for the mean-value of elta(n) over friable integers.

Improved understanding of the distribution of divisors among integers with small prime factors.

Contributed to the analytic number theory of friable integers and divisor functions.

Abstract

For integer $n$ and real $u$, define $\Delta(n,u):= |\{d : d \mid n,\,{\rm e}^u <d\leqslant {\rm e}^{u+1} \}|$. Then, put $ \Delta(n):=\max_{u\in{\mathbb R}} \Delta(n,u).$ We provide uniform upper and lower bounds for the mean-value of $\Delta(n)$ over friable integers, i.e. integers free of large prime factors.