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Cotangent power sums and character coordinates

Kurt Girstmair

Abstract

We show that certain sums studied in two recent papers are basically character coordinates (as they are called in the literature). These sums involve values of Dirichlet characters and powers of $\cot(πk/n)$, $1\le k\le n-1$. We also show that a basic tool for the study of these sums was already given in 1987, in the form of the character coordinates of so-called cotangent numbers. By means of this tool, we obtain the results of the said papers in a simple and lucid way. We also show that the coefficients of the linear combinations used in the said papers are essentially the same.

Cotangent power sums and character coordinates

Abstract

We show that certain sums studied in two recent papers are basically character coordinates (as they are called in the literature). These sums involve values of Dirichlet characters and powers of , . We also show that a basic tool for the study of these sums was already given in 1987, in the form of the character coordinates of so-called cotangent numbers. By means of this tool, we obtain the results of the said papers in a simple and lucid way. We also show that the coefficients of the linear combinations used in the said papers are essentially the same.

Paper Structure

This paper contains 4 sections, 3 theorems, 45 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Character coordinates
  3. 3. Cotangent powers and cotangent derivatives
  4. 4. Another form of the coefficients $c_{r,j}$

Key Result

Proposition 1

For $r\ge 1$, we have with $c_{r,j}$ as in (3.6).

Theorems & Definitions (3)

  • Proposition 1
  • Theorem 1
  • Theorem 2