The Riemann Zeta Function has been successfully and promisingly generalized in various ways so that the concept of zeta functions has become important in many different areas of research. In particular, work done by Y. Ihara in the 1960s led to the definition of an Ihara Zeta Function for finite graphs. The Ihara Zeta Function has the nice property of having three equivalent expressions: an Euler product form over ``primes" of the graph, an expression in terms of vertex operators on the graph, and an expression in terms of arc operators on the graph. In this paper we present two possibilities for generalizing the Ihara Zeta Function to cell products of graphs. We start with a background discussion of the Ihara Zeta Function and cell products. Then we present our generalized zeta functions and prove some properties about them. Our hope is that the ideas presented in this paper will stimulate further ideas about using the nice properties of the Ihara Zeta Function as a model for defining zeta functions more generally on higher dimensional geometric objects.

Author Bio

Zuhair Khandker is a member of the Undergraduate Class of 2007 atPrinceton University and is interested in mathematics and physics.He explored the topic of zeta functions of graphs during theSummer 2005 Mathematics REU Program at Louisiana State Universityand began this paper at that time. Outside of school, Zuhairenjoys following sports, like college football. He is lookingforward to receiving advice and comments relating to his researchand is excited about learning new ways of understanding zetafunctions of graphs and higher-dimensional generalizations ofthem.