1. Introduction

This paper is an expanded version of lectures given at MSRI in June of 2008. It provides an introduction to various zeta functions emphasizing zeta functions of a finite graph and connections with random matrix theory and quantum chaos.

For the number theorist, most zeta functions are multiplicative generating functions for something like primes (or prime ideals). The Riemann zeta is the chief example. There are analogous functions arising in other fields such as Selberg's zeta function of a Riemann surface, Ihara's zeta function of a finite connected graph. All of these are introduced in Section 2. We will consider the Riemann hypothesis for the Ihara zeta function and its connection with expander graphs.

Chapter 3 starts with the Ruelle zeta function of a dynamical system, which will be shown to be a generalization of the Ihara zeta. A determinant formula is proved for the Ihara zeta function. Then we prove the graph prime number theorem.

In Section 4 we define two more zeta functions associated to a finite graph: the edge and path zetas. Both are functions of several complex variables. Both are reciprocals of polynomials in several variables, thanks to determinant formulas. We show how to specialize the path zeta to the edge zeta and then the edge zeta to the original Ihara zeta. The Bass proof of Ihara's determinant formula for the Ihara zeta function is given. The edge zeta allows one to consider graphs with weights on the edges. This is of interest for work on quantum graphs. See [Smilansky 2007] or [Horton et al. 2006b].

Lastly we consider what the poles of the Ihara zeta have to do with the eigen-values of a random matrix. That is the sort of question considered in quantum chaos theory. Physicists have long studied spectra of Schrödinger operators and random matrices thanks to the implications for quantum mechanics where eigenvalues are viewed as energy levels of a system. Number theorists such as A.