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Reciprocity For Dedekind Sums via Conical Zeta Values

Yerko Torres-Nova

TL;DR

The work addresses reciprocity laws for Dedekind sums associated with absolutely continuous periodic functions, extending the classical Dedekind–Rademacher reciprocity and incorporating periodic Bernoulli functions. It develops integral and Fourier approaches to derive general reciprocity formulas, and ties these identities to conical zeta values via a polyhedral-geometry viewpoint that uses desingularization to unimodular cones. A two-step decomposition of conical zeta values and a detailed treatment of dimension two yield explicit reciprocity formulas that unify arithmetic, Fourier analysis, and polyhedral geometry, including a recovery of the classical case when appropriate. The results illuminate deep connections between lattice sums, Bernoulli polynomials, and cone geometry with potential implications for number theory and topology.

Abstract

We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind-Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach generalizes an integral method and uses Fourier analysis to show that the reciprocity for polynomial-type functions admits a geometric interpretation in terms of conical zeta values.

Reciprocity For Dedekind Sums via Conical Zeta Values

TL;DR

The work addresses reciprocity laws for Dedekind sums associated with absolutely continuous periodic functions, extending the classical Dedekind–Rademacher reciprocity and incorporating periodic Bernoulli functions. It develops integral and Fourier approaches to derive general reciprocity formulas, and ties these identities to conical zeta values via a polyhedral-geometry viewpoint that uses desingularization to unimodular cones. A two-step decomposition of conical zeta values and a detailed treatment of dimension two yield explicit reciprocity formulas that unify arithmetic, Fourier analysis, and polyhedral geometry, including a recovery of the classical case when appropriate. The results illuminate deep connections between lattice sums, Bernoulli polynomials, and cone geometry with potential implications for number theory and topology.

Abstract

We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind-Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach generalizes an integral method and uses Fourier analysis to show that the reciprocity for polynomial-type functions admits a geometric interpretation in terms of conical zeta values.
Paper Structure (13 sections, 13 theorems, 137 equations)

This paper contains 13 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemann-Stieltjes Integration
  4. Dedekind Sums
  5. Conical Zeta Values
  6. General Reciprocity Formulas
  7. The Integral Reciprocity
  8. Fourier Approach
  9. Reciprocity for Polynomial Dedekind Sums
  10. Zeta Values and Polyhedra
  11. The Multiple zeta values.
  12. Desingularization Algorithms for Polyhedra
  13. The Case of Dimension $2$.

Key Result

Theorem 2.1

Let $f,g,h:[a,b] \to {\mathbb C}$ be functions of bounded variation with no common discontinuities. Then,

Theorems & Definitions (25)

  • Theorem 2.1: Integration by Parts
  • Theorem 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Example 2.8
  • Theorem 3.1
  • Example 3.2
  • ...and 15 more