Fourier–Dedekind sums and an extension of Rademacher reciprocity
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 pseudorandom 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, in a precise sense, almost 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 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.”
KeywordsFourier–Dedekind sum Dedekind sum Rademacher reciprocity Lattice points
Mathematics Subject Classification11F20 05A15 11L03 52C07
It is a pleasure to acknowledge interesting discussions and valuable feedback from M. Beck, D. Cristofaro-Gardiner, K. Ribet, S. Robins, J. Sondow, and X. Yuan. This work originated during the 2013 UC Berkeley Geometry, Topology, and Operator Algebras RTG Summer Research Program for Undergraduates. This program was funded by the National Science Foundation.