A lower bound on the mean value of the Erdős-Hooley Delta function
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916256216514560 |
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| author | Ford, Kevin Koukoulopoulos, Dimitris Tao, Terence |
| author_facet | Ford, Kevin Koukoulopoulos, Dimitris Tao, Terence |
| contents | We give an improved lower bound for the average of the Erdős-Hooley function $Δ(n)$, namely $\sum_{n\le x} Δ(n) \gg_\varepsilon x(\log\log x)^{1+η-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $η= 0.3533227\dots$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11987 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A lower bound on the mean value of the Erdős-Hooley Delta function Ford, Kevin Koukoulopoulos, Dimitris Tao, Terence Number Theory Primary: 11N25, Secondary: 11N37, 11N64 We give an improved lower bound for the average of the Erdős-Hooley function $Δ(n)$, namely $\sum_{n\le x} Δ(n) \gg_\varepsilon x(\log\log x)^{1+η-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $η= 0.3533227\dots$ 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. |
| title | A lower bound on the mean value of the Erdős-Hooley Delta function |
| topic | Number Theory Primary: 11N25, Secondary: 11N37, 11N64 |
| url | https://arxiv.org/abs/2308.11987 |