Average sizes of mixed character sums
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915756800737280 |
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| author | Wang, Victor Y. Xu, Max Wenqiang |
| author_facet | Wang, Victor Y. Xu, Max Wenqiang |
| contents | We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} χ(n) e(nθ) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $θ$ satisfying a weak Diophantine condition, where $χ$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $θ$. Certain quadratic Diophantine equations play a key role in the present paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average sizes of mixed character sums Wang, Victor Y. Xu, Max Wenqiang Number Theory Probability We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} χ(n) e(nθ) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $θ$ satisfying a weak Diophantine condition, where $χ$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $θ$. Certain quadratic Diophantine equations play a key role in the present paper. |
| title | Average sizes of mixed character sums |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2411.14181 |