A Zeta Function for Juggling Sequences

Abstract

We give a new generalization of the Riemann zeta function to the set of b-ball juggling sequences. We develop several properties of this zeta function, noting among other things that it is rational in b-s. We provide a meromorphic continuation of the juggling zeta function to the entire complex plane (except for a countable set of singularities) and completely enumerate its zeroes. For most values of b, we are able to show that the zeroes of the b-ball zeta function are located within a critical strip, which is closely analogous to that of the Riemann zeta function.