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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.03347 |
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| _version_ | 1866910224350183424 |
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| author | Liu, Jianya Xie, Sizhe |
| author_facet | Liu, Jianya Xie, Sizhe |
| contents | A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \ldots, F_R \in \mathbb{Z}[x_1, \ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\boldsymbol{F} = (F_1, \ldots, F_R)$ be nonsingular. We prove that the system $\boldsymbol{F}(\boldsymbol{x})=\mathbf{0}$ is solvable in primes provided that $s \geq D^2 4^{D+2} R^5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiple Gauss sums Liu, Jianya Xie, Sizhe Number Theory A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \ldots, F_R \in \mathbb{Z}[x_1, \ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\boldsymbol{F} = (F_1, \ldots, F_R)$ be nonsingular. We prove that the system $\boldsymbol{F}(\boldsymbol{x})=\mathbf{0}$ is solvable in primes provided that $s \geq D^2 4^{D+2} R^5$. |
| title | Multiple Gauss sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.03347 |