Trigonometric Determinants via special values of Dirichlet $L$-Functions

TL;DR

This paper explores the relationship between trigonometric determinants and special values of Dirichlet L-functions, extending previous results and proving a conjecture, using spectral decomposition methods.

Contribution

It establishes a new connection between trigonometric determinants and Dirichlet L-functions, generalizes Guo's results, and proves Zhi-Wei Sun's conjecture.

Findings

Derived explicit formulas for sine matrix determinants.

Connected determinants with Dirichlet L-function values.

Proved a conjecture by Zhi-Wei Sun.

Abstract

In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results to arbitrary positive integers n. In addition, we also prove a conjecture raised by Zhi-Wei Sun. Our main tool is the spectral decomposition of some linear operators. By the same method we obtain an explicit formula for the determinants of sine matrices. This formula is expressed as a product of Gauss sums attached to Dirichlet characters.