### Generating Functions and Generalized Dedekind Sums

#### Abstract

We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ 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 $\prod_\zeta (1-xR(\zeta))$. *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.