Generating Functions and Generalized Dedekind Sums

We study sums of the form _ R (), where R is a rational function and the sum is over all nth roots of unity (often with =1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for _ (1-xR () ). Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.