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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.14795 |
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| _version_ | 1866913046873505792 |
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| author | Liu, Hang |
| author_facet | Liu, Hang |
| contents | In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup $\mathbb{Z}/N\mathbb{Z}$. We show that, via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group determinants indexed by $d|N$. This yields explicit formulas for several determinant families, including determinants involving periodic Bernoulli functions and powers of the tangent function. As an application, we also prove a corrected version of a conjecture of Sun Zhi-Wei. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14795 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform Liu, Hang Number Theory 11C20, 11B68, 11M06, 20M25 In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup $\mathbb{Z}/N\mathbb{Z}$. We show that, via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group determinants indexed by $d|N$. This yields explicit formulas for several determinant families, including determinants involving periodic Bernoulli functions and powers of the tangent function. As an application, we also prove a corrected version of a conjecture of Sun Zhi-Wei. |
| title | Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform |
| topic | Number Theory 11C20, 11B68, 11M06, 20M25 |
| url | https://arxiv.org/abs/2603.14795 |