Trigonometric representations of generalized Dedekind and Hardy sums via the discrete Fourier transform

TL;DR

This paper introduces higher-dimensional generalizations of Dedekind and Hardy sums using Bernoulli functions and sawtooth functions, deriving finite trigonometric representations via a generalized Parseval's formula for the discrete Fourier transform.

Contribution

It presents new higher-dimensional generalizations of Dedekind and Hardy sums and a unified method for their trigonometric representations using Fourier analysis.

Findings

Derived finite trigonometric representations for generalized sums

Extended Parseval's formula for the discrete Fourier transform

Connected sums involving the Hurwitz zeta function

Abstract

We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the discrete Fourier transform to derive finite trigonometric representations for these sums in a simple unified manner. We also consider a related sum involving the Hurwitz zeta function.