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Hecke's inverse cotangent numbers

Kurt Girstmair

Abstract

In connection with Eisenstein series for the principal congruence subgroup $Γ(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the $n$th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that $n$ is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums.

Hecke's inverse cotangent numbers

Abstract

In connection with Eisenstein series for the principal congruence subgroup , Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums.

Paper Structure

This paper contains 6 sections, 8 theorems, 74 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Inverse cotangent numbers
  3. 3. Generalized Bernoulli matrices
  4. 4. Some examples
  5. 5. Series giving rational values
  6. 6. Eigenvalues and character coordinates

Key Result

Proposition 1

Let $n$ and $r$ be as above. Let ${\cal R}=\{j; 1\le j\le n/2, (j,n)=1\}.$ For a number $j\in\mathbb Z$, $(j,n)=1$, let $j^*$ denote the inverse of $j$ mod $n$, i.e., $jj^*\equiv 1\mod n$, $1\le j^*\le n$. If ${\rm{ct}}^{(r)}_j$ and $\widehat{\rm{ct}}^{(r)}_j$ are defined by (1.2), (1.4), and (1.6), for all $s,t\in{\cal R}$. Here $\delta$ is the Kronecker delta.

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Proposition 4
  • Theorem 1
  • Proposition 5
  • Theorem 2