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Higher Weight Generalized Dedekind Sums

Preston Tranbarger

TL;DR

This work extends the classical Dedekind-sum framework by defining higher-weight generalized Dedekind sums S_{χ1,χ2,k} as period integrals of weight k holomorphic Eisenstein series attached to primitive Dirichlet characters. It establishes a finite-sum formula, proves quantum-modular behavior of the corrected cusp-cusp sums, and develops a cohomological and analytic structure, including an independent-point formula and a nontrivial upper bound. The paper also investigates arithmetic properties of the sums’ images, Fricke reciprocity, and provides extensive computational evidence for higher weights, including a containment conjecture for the image and explicit computations up to weight 9 and small conductors. Together, these results generalize SVY’s k=2 theory to arbitrary weight, linking Bernoulli character polynomials, period integrals, and modular/quantum-modular phenomena with concrete computational data. The methods blend analytic, algebraic, and computational techniques, offering both theoretical insight and practical tools for studying higher-weight Dedekind sums and their arithmetic implications.

Abstract

Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we explore a variety of properties of these generalized Dedekind sums: we develop a finite sum formula, demonstrate their behavior as quantum modular forms, provide a Fricke reciprocity law, and characterize analytic and arithmetic aspects of their image. Particularly, for the arithmetic aspect of the image, we generalize an existing conjecture to the higher weight case and provide significant computational evidence to support this generalized conjecture.

Higher Weight Generalized Dedekind Sums

TL;DR

This work extends the classical Dedekind-sum framework by defining higher-weight generalized Dedekind sums S_{χ1,χ2,k} as period integrals of weight k holomorphic Eisenstein series attached to primitive Dirichlet characters. It establishes a finite-sum formula, proves quantum-modular behavior of the corrected cusp-cusp sums, and develops a cohomological and analytic structure, including an independent-point formula and a nontrivial upper bound. The paper also investigates arithmetic properties of the sums’ images, Fricke reciprocity, and provides extensive computational evidence for higher weights, including a containment conjecture for the image and explicit computations up to weight 9 and small conductors. Together, these results generalize SVY’s k=2 theory to arbitrary weight, linking Bernoulli character polynomials, period integrals, and modular/quantum-modular phenomena with concrete computational data. The methods blend analytic, algebraic, and computational techniques, offering both theoretical insight and practical tools for studying higher-weight Dedekind sums and their arithmetic implications.

Abstract

Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we explore a variety of properties of these generalized Dedekind sums: we develop a finite sum formula, demonstrate their behavior as quantum modular forms, provide a Fricke reciprocity law, and characterize analytic and arithmetic aspects of their image. Particularly, for the arithmetic aspect of the image, we generalize an existing conjecture to the higher weight case and provide significant computational evidence to support this generalized conjecture.

Paper Structure

This paper contains 23 sections, 40 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction and Basic Definitions
  2. Finite Sum Formula
  3. Quantum Modularity
  4. Arithmetic Aspects of $S_{\chi_1,\chi_2,k}(\Gamma_1(q_1q_2))$
  5. Fricke Reciprocity
  6. Analytic Aspects of $S_{\chi_1,\chi_2,k}(\Gamma_0(q_1q_2))$
  7. $S_{\chi_1,\chi_2,k}$ as a Finite Sum
  8. Results from the Literature
  9. Analysis Preliminaries
  10. Proof of Theorem \ref{['mainThm']}
  11. Cohomological Properties of $\phi_{\chi_1,\chi_2,k}$ and $S_{\chi_1,\chi_2,2}$
  12. A Non-Trivial Upper Bound on $\abs{S_{\chi_1,\chi_2,k}}$
  13. Higher Weight Independent Point Formula
  14. Literature Results for the Proof of Theorem \ref{['nontrivialbound']}
  15. Proof of Theorem \ref{['nontrivialbound']}
  16. ...and 8 more sections

Key Result

Lemma 1.3

We have the following Fourier series for $E_{\chi_1,\chi_2,k}$:

Figures (3)

  • Figure 1: Scatter plot of $\widehat{S}_{\chi_1,\chi_2,k}(\mathcal{C}_{50}(q_1q_2))$
  • Figure 2: Scatter plots of $h_{\gamma,\chi_1,\chi_2,k}(\mathcal{C}_{15}(q_1q_2))$ for two selected values of $\gamma\in\Gamma_1(q_1q_2)$
  • Figure 3: Scatter plots of $h_{\gamma,\chi_1,\chi_2,k}(\mathcal{C}_{15}(q_1q_2))$ for $\gamma=\gamma_1\gamma_2$ and $\gamma=\gamma_2\gamma_1$.

Theorems & Definitions (98)

  • Definition 1.1
  • Definition 1.2
  • Remark
  • Lemma 1.3
  • Remark
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • ...and 88 more