Dedekind Sums, the Building Blocks of Lattice-Point Enumeration

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We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, 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 shall 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 \(\log _{2}(b) = 100\) steps in this example. This is the kind of magic that saves the day when we try to enumerate lattice points in integral polytopes of dimension d ≤ 4. In this chapter, we focus on the computational-complexity issues that arise when we try to compute Dedekind sums explicitly. In many ways, the Dedekind sums extend the notion of the greatest common divisor of two integers.