Reciprocity For Dedekind Sums via Conical Zeta Values

TL;DR

This paper extends classical reciprocity formulas for Dedekind sums, especially for periodic Bernoulli functions, using Fourier analysis and conical zeta values to provide a geometric interpretation.

Contribution

It introduces a generalized approach to reciprocity formulas for Dedekind sums via conical zeta values and Fourier analysis, expanding the classical theory.

Findings

Reciprocity formulas for Dedekind sums are extended to absolutely continuous functions.

A geometric interpretation of reciprocity for polynomial-type functions is provided.

The approach unifies integral methods with Fourier analysis and conical zeta values.

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.