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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.17778 |
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| _version_ | 1866909917090152448 |
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| author | Hasanalizade, Elchin Lin, Hua Martin, Greg Martínez, Andradis Luna Treviño, Enrique |
| author_facet | Hasanalizade, Elchin Lin, Hua Martin, Greg Martínez, Andradis Luna Treviño, Enrique |
| contents | Burgess proved that for $χ_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\Big|\sum_{M< n\le M+N}χ_q(n)\Big| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+ε}$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Burgess inequalities for cubefree moduli Hasanalizade, Elchin Lin, Hua Martin, Greg Martínez, Andradis Luna Treviño, Enrique Number Theory 11L40 Primary, 11Y60 Secondary Burgess proved that for $χ_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\Big|\sum_{M< n\le M+N}χ_q(n)\Big| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+ε}$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$. |
| title | Explicit Burgess inequalities for cubefree moduli |
| topic | Number Theory 11L40 Primary, 11Y60 Secondary |
| url | https://arxiv.org/abs/2511.17778 |