# What are zeta functions good for?

I know a couple of answers to the above question:

1. They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.

2. There are various conjectures/results relating the special values of L-functions with other stuff in the vein of the class number formula/Birch Swinnerton-Dyer conjecture, Iwasawa theory on the other hand.

What else can we do (conjecturally or otherwise) with zeta functions? I am interested in connections of the zeta functions to objects that have nothing to do with zeta functions as such (but are still of interest to arithmetic geometers and other mathematicians). My interests and background are definitely very algebraic so I have almost no idea about what results on the analytic side imply.

• See the book "Non-vanishing of $L$-functions and Applications" by Murty and Marty. There are a lot of applications of the Generalized Riemann Hypothesis (once it is proved) and some of these have later been proved unconditionally by other methods. A list of some are at mathoverflow.net/questions/17209/… – KConrad Apr 1 '18 at 13:19
• – Tom Copeland Apr 1 '18 at 19:18

## 2 Answers

Zeta-function regularization is a powerful method in spectral theory, with many applications in physics, including the Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking, and non-commutative spacetime. See Ten Physical Applications of Spectral Zeta Functions, or these lecture notes:

It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose-Einstein condensation. We will consider exclusively spectral zeta functions, that is zeta functions arising from the eigenvalue spectrum of suitable differential operators. There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every spectral zeta function. Those tools are introduced using the well-studied examples of the Hurwitz, Epstein and Barnes zeta function. It is explained how these different examples of zeta functions can all be thought of as being generated by the same mechanism, namely they all result from eigenvalues of suitable (partial) differential operators. Motivations come from the questions "Can one hear the shape of a drum?" and "What does the Casimir effect know about a boundary?". Finally "What does a Bose gas know about its container?"

Depending on what kind of zeta functions you want, the Selberg zeta function allows you to relate lengths of closed geodesics to eigenvalues of the Laplacian. In particular, you can use the Selberg zeta function in combination with a trace formula to prove the prime geodesic theorem for compact Riemann surfaces and get Weyl's law. This also leads to construct isospectral manifolds.

Similarly, for graphs one can look at the analogous Ihara zeta function to relate lengths of "geodesics" to certain spectral quantities. In particular, one can get a characterization of Ramanujan graphs in terms of the Ihara zeta function. There are also numerous variants to count different things in graphs (Bartholdi zeta function, path zeta functions), and I have a conjecture with Christina Durfee that zeta functions are better at distinguishing graphs spectrally than the usual (adjacency matrix or Laplacian) spectra considered.