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

TL;DR

This paper explores paratrophic determinants over modular integers, revealing their factorization via Fourier transforms and providing explicit formulas, with applications to conjectures involving Bernoulli functions and tangent powers.

Contribution

It introduces a novel factorization approach for paratrophic determinants using Fourier transforms and corrects a conjecture by Sun Zhi-Wei.

Findings

Determinants factor into products indexed by divisors of N.

Explicit formulas derived for determinants involving Bernoulli functions.

A corrected version of Sun Zhi-Wei's conjecture is proved.

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.