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The equidistribution of Elliptic Dedekind sums and generalized Selberg-Kloosterman sums

Kim Klinger-Logan, Tian An Wong

Abstract

We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg-Kloosterman sums for discrete subgroups of $\PSL_2(\mathbb C)$ using Poincaré series.

The equidistribution of Elliptic Dedekind sums and generalized Selberg-Kloosterman sums

Abstract

We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg-Kloosterman sums for discrete subgroups of using Poincaré series.
Paper Structure (13 sections, 7 theorems, 83 equations)

This paper contains 13 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The distribution of elliptic Dedekind sums
  3. Sums of generalized Selberg-Kloosterman sums
  4. Definitions
  5. Upper Half-Space
  6. Eisenstein series
  7. Poincaré series
  8. Bounds for generalized Selberg-Kloosterman sums
  9. Preliminary estimates
  10. Proof of Theorem \ref{['thm:GSthm2']}
  11. Proof of equidistribution
  12. Coset counting
  13. Proof of Theorem 1

Key Result

Theorem 1

For all positive $r\in\mathbb R$, the sequence is equidistributed on $[0,1)$ as $|c|\to\infty$.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 3 more