Fourier-Dedekind Sums and an Extension of Rademacher Reciprocity

TL;DR

This paper explores Fourier-Dedekind sums, extending their reciprocity law, analyzing their properties, average behavior, and extrema, with applications in number theory and related fields.

Contribution

It introduces new properties, extends Rademacher reciprocity, and analyzes extrema of Fourier-Dedekind sums, providing a deeper understanding of their structure and behavior.

Findings

Fourier-Dedekind sums can be expressed as convolutions of simpler sums

Extended Rademacher reciprocity to a broader range of sums

Established bounds and extrema for 2-dimensional Fourier-Dedekind sums

Abstract

Fourier-Dedekind sums are a generalization of Dedekind sums - important number-theoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds and pseudo random number generators. A remarkable feature of Fourier-Dedekind sums is that they satisfy a reciprocity law called Rademacher reciprocity. In this paper, we study several aspects of Fourier-Dedekind sums: properties of general Fourier-Dedekind sums, extensions of the reciprocity law, average behavior of Fourier-Dedekind sums, and finally, extrema of 2-dimensional Fourier-Dedekind sums. On properties of general Fourier-Dedekind sums we show that a general Fourier-Dedekind sum is simultaneously a convolution of simpler Fourier-Dedekind sums, and a linear combination of these with integer coefficients. We show that Fourier-Dedekind sums can be extended naturally to a group under convolution. We introduce "Reduced Fourier-Dedekind sums", which encapsulate the complexity of a Fourier-Dedekind sum, describe these in terms of generating functions, and give a geometric interpretation. Next, by finding interrelations among Fourier-Dedekind sums, we extend the range on which Rademacher reciprocity Theorem holds. We go on to study the average behavior of Fourier-Dedekind sums, showing that the average behavior of a Fourier-Dedekind sum is described concisely by a lower-dimensional, simpler Fourier-Dedekind sum. Finally, we focus our study on 2-dimensional Fourier-Dedekind sums. We find tight upper and lower bounds on these for a fixed $t$, estimates on the argmax and argmin, and bounds on the sum of their "reciprocals".