Integrality of a trigonometric determinant arising from a conjecture of Sun
Liwen Gao, Xuejun Guo
TL;DR
This paper resolves a conjecture of Zhi-Wei Sun on the integrality of trigonometric determinants by establishing a new framework that relates trigonometric determinant matrices to Dirichlet $L$-functions via a parity correspondence between odd Dirichlet characters modulo $4n$ and even characters modulo $n$. The authors derive a main determinant formula for odd $n\ge3$ with $m=\phi(n)/2$, $\det(D_n)=\varepsilon(n)(-1)^{\frac{(n-1)m}{2}}\left(\frac{2n}{\pi}\right)^m\prod_{\chi_n\text{even}}\chi_n(4)L(1,\chi_n\psi)$, and show how this reduces to explicit expressions in terms of class numbers $h_p^-$ and $h_{4p}^-$ for primes, thereby refining Sun's conjecture. They provide corollaries for the prime case and demonstrate that Sun's original formulation is not universally valid, while the new formula reveals a structured arithmetic underpinning of these determinants. The work highlights deep connections between Fourier-analytic matrix structures and arithmetic invariants, suggesting potential applicability to other classes of structured matrices with symmetry arising from Dirichlet characters and $L$-functions.
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.