Zeta functions are of interest primarily because they {\it count} things. Thus, they are basic combinatorial objects that encode information (number-theoretic, geometric, graph-theoretic, or even topological). To a purist, in order to qualify for the name {\it zeta function,} such a counting function is supposed to satisfy a functional equation (the values at positive arguments are related in a precise way to the values at negative arguments) and have an expansion as an Euler product; however, the name ``zeta" seems to be spreading these days even when some of these features are missing. The original zeta function is the Riemann zeta function; today there are zeta functions of algebraic number fields, zeta functions of elliptic curves, zeta functions of Riemannian manifolds, zeta functions of modular forms, and there are zeta functions of graphs. The appeal of the topic is that the name `zeta function' is important in connection with Fermat's Last Theorem, via the Taniyama-Shimura conjecture (now a theorem), or in connection with the Riemann Hypothesis. The appeal for a summer REU is that there are certain aspects of zeta functions that are entirely elementary and accessible to direct investigation. This means that one can get quickly involved with making examples--by hand or by machine computation--while the deeper connections invite the student to delve into higher mathematics. A Focus Question for this component of the REU, stated for the purpose of fixing ideas: \textit{Can one construct two nonisomorphic but isospectral Ramanujan graphs}? ``Isospectral'' in this context means that the adjacency matrices of the graphs should have the same eigenvalues. The Ihara-Selberg zeta function of the graphs are built out of these eigenvalues, and the question can be reformulated: Can one construct two nonisomorphic Ramanujan graphs having identical Ihara-Selberg zeta functions? This question is quite current: see the works of Bass \cite{B}, of Langlands-Chung \cite{CL}, of Jordan-Livne \cite{JL}, and of Li \cite{WL}.
Students will become involved by means of permutation representations of finite groups. This is a topic accessible to anyone having had an undergraduate course in algebra, and is basically the notion of groups acting on sets. When a finite group $G$ acts (transitively) on a set $S$, then for each element $g\in G$, the {\it permutation character, $\chi _S (g)$,} counts the fixed points of $g.$ The phenomenon of interest is that it is possible for there to be two non-isomorphic (transitive) $G$-sets $S$ and $S'$ for which $$\chi _S(g) = \chi _{S'}(g)$$ for all $g\in G$. So in terms of fixed-points, the sets $S$ and $S'$ are indistinguishable. This phenomenon was discovered in 1926 by F. Gassmann. The surprise is this: Once $G$ is be realized as the Galois group of some normal extension $N|Q,$ these two different permutation representations of $G$ give rise to two nonisomorphic algebraic number fields $K, K'$ having identical Dedekind zeta functions (these zeta functions being precisely the Artin L-series built out of the two permutation representations in question). Such pairs of number fields $K, K'$ are called {\it arithmetically equivalent}. There is even a group $G$ of order 32 (the group the semi-direct product of $(Z/8Z)^*$ acting as automorphisms on the cyclic group $Z/8Z$) that allows two such permutation representations, and students can study permutation actions of this group by direct hand computation. The interesting open question was to compare the class numbers of two arithmetically equivalent fields. In 1994, de Smit and Perlis gave the first examples of arithmetically equivalent fields whose class numbers differ by a factor of 2. Four years later, de Smit gave another example of arithmetically equivalent fields whose class numbers differ by a factor of 3, and gave some heuristics to indicate that one might not easily find a similar example with class numbers differing by a factor of 5. However, the 1999 LSU REU student G. Dyer found exactly such an example by a computer search mimicking de Smit's earlier work.
In 1985 T. Sunada came up with a way of constructing two nonisometric Riemannian manifolds that are isospectral, meaning that the eigenvalues of the Laplace operators on the two manifolds agree. Since these eigenvalues represent the harmonics of a vibrating surface, these manifolds are negative answers to M. Kac's famous query ``Can you hear the shape of a drum?" (Here, the drum does not have a fixed boundary). These examples have been studied by Robert Brooks, Carolyn Gordon, Dennis de Turck, and especially by Peter Buser and others to the point where constructing such manifolds is a matter of cutting and pasting with ordinary scissors and paper. One can build a Dirichlet series out of the eigenvalues of the Laplacian on the manifolds, and some authors have taken to calling these Dirichlet series ``zeta functions" of the manifolds. So these isospectral manifolds are the exact analog of arithmetically equivalent number fields. The Focus Question for this component of the summer REU is to try to adapt these same methods to the Ihara-Selberg zeta functions of Ramanujan graphs.
The zeta function of a graph can be defined for any finite graph in terms of the eigenvalues of its adjacency matrix (\cite {B} provides an introduction to this). Graphs whose eigenvalues satisfy certain bounds are called Ramanujan graphs, a notion introduced by Lubotzky, Phillips and Sarnak, \cite{LPS}. Ramanujan graphs are important in communication and number theory. The spectral theory of finite graphs has become a subject in its own right (for a sampling of recent works, see \cite{C}, \cite{CL}, \cite{JL}). There are now several elementary methods known for constructing Ramanujan graphs (although the proofs of the eigenvalue bounds are highly nontrivial). For instance, Winnie Li has has given constructions of Ramanujan graphs based on both abelian and nonabelian groups (see chapter 9 of \cite{WL} and the references therein). It is certainly not known whether or not the permutation representation construction that gives {\it all} examples of arithmetically equivalent number fields and also gives {\it many} examples of nonisometric but isospectral Riemannian manifolds can be modified to give a construction of isometric Ramanujan graphs. The topic is exciting, and so accessible, and the payoff is potentially big so that it is appropriately chosen as the Focus Question for this component of the REU.