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arXiv · 2603.14795

Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform

Hang Liu
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Abstract

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.

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Hang Liu. 2026-03-16. Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform. https://arxiv.org/abs/2603.14795

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