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

Liwen Gao, Xuejun Guo

TL;DR

The paper reveals a deep link between determinants of trigonometric matrices and special values of Dirichlet L-functions, extending Guo’s results to all positive integers n and proving Sun’s conjectures (with refinements). Using a spectral decomposition framework, it derives explicit determinant formulas for cotangent, tangent, cosecant, and sine matrices in terms of L(1, χ), Gauss sums, and related arithmetic invariants such as class numbers h_n^- and conductors. The results unify several strands of analytic and algebraic number theory, providing exact, computable expressions and illuminating number-theoretic conditions under which these determinants vanish or take particular values. This work enhances the understanding of how finite-trigonometric determinants encode deep arithmetic data and offers practical formulas for applications in computation and SEO-rich mathematical context.

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.

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

TL;DR

The paper reveals a deep link between determinants of trigonometric matrices and special values of Dirichlet L-functions, extending Guo’s results to all positive integers n and proving Sun’s conjectures (with refinements). Using a spectral decomposition framework, it derives explicit determinant formulas for cotangent, tangent, cosecant, and sine matrices in terms of L(1, χ), Gauss sums, and related arithmetic invariants such as class numbers h_n^- and conductors. The results unify several strands of analytic and algebraic number theory, providing exact, computable expressions and illuminating number-theoretic conditions under which these determinants vanish or take particular values. This work enhances the understanding of how finite-trigonometric determinants encode deep arithmetic data and offers practical formulas for applications in computation and SEO-rich mathematical context.

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.

Paper Structure

This paper contains 5 sections, 21 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Determinants of cotangent matrices
  3. Determinants of tangent matrices
  4. Determinants of cosecant matrices
  5. Determinants of sine matrices

Key Result

Theorem 1.4

Let $n\geq 3$ be a positive integer. Then Equivalently, an explicit form convenient for computation is where $\chi^*$ stands for the primitive character associated with $\chi$ in the sense of Lemma char, $w$ is the number of roots of unity in $\mathbb{Q}(\zeta_n)$, $f_{\chi}$ is the conductor of the character $\chi$, $h_n^-$ denotes the relative class number of $\mathbb{Q}(\zeta_n)$ and Moreove

Theorems & Definitions (43)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 33 more