A lower bound on the mean value of the Erd\H{o}s-Hooley Delta function
| Autor: | Ford, Kevin, Koukoulopoulos, Dimitris, Tao, Terence |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | We give an improved lower bound for the average of the Erd\H{o}s-Hooley function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and third authors. Comment: 15 pages. Added remarks in the end of page 3; added references [3] and [9]. Final version, to appear in Proc. London Math. Soc |
| Databáze: | arXiv |
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