Dedekind Sums, the Building Blocks of Lattice-point Enumeration

We’ve encountered Dedekind sums in our study of finite Fourier analysis and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1. They have one shortcoming, however (which we'll remove): the definition of s(a, b) requires us to sum over b terms, which is rather slow when b = 2¹⁰⁰, for example. Luckily, there is a magical reciprocity law for the Dedekind sum s(a, b) that allows us to compute it in roughly log₂ (b) = 100 steps. This is the kind of magic that saves the day when we try to enumerate lattice points in integral polytopes of dimensions d≤4. There is an ongoing effort to extend these ideas to higher dimensions, but there is much room for improvement. In this chapter we focus on the computational-complexity issues that arise when we try to compute Dedekind sums explicitely.