Integrality of a trigonometric determinant arising from a conjecture of Sun

TL;DR

This paper proves a conjecture by Sun on the integrality of specific trigonometric determinants, revealing deep links between Fourier analysis, number theory, and arithmetic structures.

Contribution

It introduces a novel approach connecting characters modulo different integers to establish integrality of trigonometric determinants.

Findings

Confirmed Sun's conjecture on integrality.

Established a new link between characters and determinants.

Enhanced understanding of arithmetic structures in trigonometric matrices.

Abstract

In this paper we resolve a conjecture of Zhi-Wei Sun concerning the integrality and arithmetic structure of certain trigonometric determinants. Our approach builds on techniques developed in our previous work, where trigonometric determinants were studied via special values of Dirichlet $L$-functions. The method is refined by establishing a connection between odd characters modulo $4n$ and even characters modulo $n$. The results highlight a close connection between trigonometric determinant matrices, Fourier-analytic structures, and arithmetic invariants.