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Note on the mean value of the Erd\H{o}s--Hooley Delta-function by de la Bretèche, Régis; Tenenbaum, Gérald is a scholarly article available to read on EtoBox.

What is Note on the mean value of the Erd\H{o}s--Hooley Delta-function about?

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 $$\sum_{n\leqslant x}\Delta(n)\ll x(\log_2x)^{ 5/2}.$$

Author

de la Bretèche, Régis; Tenenbaum, Gérald

Published

2023

Language

EN