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arXiv · 2309.03958

Note on the mean value of 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 a recent upper bound for the mean value of this function by showing that, for large $x$, we have $$\sum_{n\leqslant x}\Delta(n)\ll x(\log_2x)^{ 5/2}.$$

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Régis de la Bretèche, Gérald Tenenbaum. 2023-09-07. Note on the mean value of the Erd\H{o}s--Hooley Delta-function. https://arxiv.org/abs/2309.03958

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