Multiple exponential sums and their applications to quadratic congruences
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909846418227200 |
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| author | Bag, Nilanjan Baier, Stephan Haldar, Anup |
| author_facet | Bag, Nilanjan Baier, Stephan Haldar, Anup |
| contents | In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistribution result for these solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_17706 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multiple exponential sums and their applications to quadratic congruences Bag, Nilanjan Baier, Stephan Haldar, Anup Number Theory 11L40, 11T23, 11K36 In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistribution result for these solutions. |
| title | Multiple exponential sums and their applications to quadratic congruences |
| topic | Number Theory 11L40, 11T23, 11K36 |
| url | https://arxiv.org/abs/2311.17706 |