Fourier-Dedekind Sums and an Extension of Rademacher Reciprocity

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".