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

Hang Liu

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. The idea of using discrete Fourier transform (DFT) in our approach was discovered through human-AI interaction.

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

Abstract

In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup . We show that, via discrete Fourier, cosine and sine transforms, these determinants factor into products of group determinants indexed by . 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. The idea of using discrete Fourier transform (DFT) in our approach was discovered through human-AI interaction.
Paper Structure (8 sections, 9 theorems, 87 equations)

This paper contains 8 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Paratrophic determinants attached to $\mathbb Z/N\mathbb Z$
  3. The matrices $X_N$, $Y_N$, and $Z_N$
  4. Fourier, cosine and sine transforms
  5. Applications of theorem \ref{['thm:det-XYZ-general']}
  6. Determinant of Bernoulli matrices
  7. Determinant of tangent matrix and a conjecture of Sun
  8. Acknowledgements

Key Result

Lemma 2.1

Let $m$ and $k$ be row and column indices of matrices and put $d:=\gcd(k,N)$. If $d\nmid m$, then If $d | m$, denote $m=du$, $k=dv$, then $\gcd(v,N_d)=1$ and where $v^{-1}$ is the inverse of $v$ modulo $N_d$.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.8
  • proof
  • Lemma 3.1
  • ...and 12 more