Products involving the real parts of Jacobi sums and related cyclotomic matrices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918529413939200 |
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| author | Wu, Hai-Liang Ji, Xiao-Han |
| author_facet | Wu, Hai-Liang Ji, Xiao-Han |
| contents | Let $q$ be an odd prime power and $χ_q$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. In this paper, we study the arithmetic properties of the product
$$R_q(χ_q)=\prod_{0<k<(q-1)/4}\left(J_q(ϕ_q,χ_q^k)+J_q(ϕ_q,χ_q^{-k})\right),$$
which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix
$$\left[ϕ_q(s_i+s_j)\right]_{1\le i,j\le (q-1)/2},$$
where $ϕ_q$ is the unique quadratic multiplicative character of $\mathbb{F}_q$, and $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Products involving the real parts of Jacobi sums and related cyclotomic matrices Wu, Hai-Liang Ji, Xiao-Han Number Theory Let $q$ be an odd prime power and $χ_q$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. In this paper, we study the arithmetic properties of the product $$R_q(χ_q)=\prod_{0<k<(q-1)/4}\left(J_q(ϕ_q,χ_q^k)+J_q(ϕ_q,χ_q^{-k})\right),$$ which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix $$\left[ϕ_q(s_i+s_j)\right]_{1\le i,j\le (q-1)/2},$$ where $ϕ_q$ is the unique quadratic multiplicative character of $\mathbb{F}_q$, and $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$. |
| title | Products involving the real parts of Jacobi sums and related cyclotomic matrices |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.27169 |