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

Régis de la Bretèche; Gérald Tenenbaum

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arXiv · 2210.13897

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

Régis de la Bretèche·Gérald Tenenbaum

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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.

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Régis de la Bretèche, Gérald Tenenbaum. 2022-10-25. Two upper bounds for the Erd\H{o}s--Hooley Delta-function. https://arxiv.org/abs/2210.13897

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