Sample RMT and Number Theory projects

Project 1:

Background: The Riemann hypothesis implies that the Riemann zeta function does not get too large on the critical line. In particular, the Riemann Hypothesis implies that $|\zeta(\tfrac12+it)|$ grows more slowly than any power of~$t$. Exactly how large it can be is an unsolved problem, and only recently was there a precise conjecture for this.

Looking at a graph of $|\zeta(\tfrac12+it)|$ strongly suggests a direct relationship between the large values of $|\zeta(\tfrac12+it)|$ and large gaps between zeros of the zeta function. That is, large values of $|\zeta(\tfrac12+it)|$ can only occur when there is a large gap between zeros of the zeta function. As far as I know, the exact relationship is unknown.

What to do first: This project can be started in terms of the zeta function or in terms of random matrices. In either case, the best first step is to plot a bunch of examples to get a feel for the problem.

For the zeta function, it would be useful to tabulate large gaps between zeros of the zeta function, along with how large the zeta function gets between the large gaps. (One can write a short program to find zeros of the zeta function, or use lists of zeros which can be found on the web. Then use Sage or another computer package to evaluate the zeta function).

For random matrices, one can generate random unitary matrices, find their eigenvalues, look for large gaps, and see how large (on the unit circle) the characteristic polynomial gets between those eigenvalues. It is likely that this data will look just like the zeta function data, but it would be good to check it just to make sure.

Given this data, one can plot it to get an idea of the relationship between large gaps and large values. The data is likely to be fairly noisy. This is probably because the size of nearby gaps also has an influence: a large gap surrounded by small gaps will have different behavior than a large gap surrounded by medium-size gaps. (See Project 2 for more about this.) So a second step could be to filter the data according to such criteria.

This experimentation will give an idea of the answer, but a different approach is needed to give a proof. One can write down an expression for the expected size of the characteristic polynomial in a gap of fixed size, although it is not clear whether or not there is hope of evaluating the resulting integral.

Possible outcomes: Even if it is not possible to prove an exact relationship between large gaps and large values, one may be able to fit to the data and conjecture the relationship. Such a conjecture would be interesting because there is a conjecture for the size of the largest gaps between zeros of the zeta function, and there is a conjecture for the size of the largest values of the zeta function. Are those two conjectures consistent with your conjectured formula for the relationship between gap size and zeta function size? I wouldn't be surprised if the results are inconsistent. That would suggest that the large values of the zeta function are even more mysterious than we realized.

Project 2:

1. Given a large gap between two eigenvalues, where do you expect the other eigenvalues to be?
2. Given an eigenvalue of high multiplicity, what is the most likely configuration of the other eigenvalues?

Background: Project 1 described motivation for understanding large gaps between eigenvalues, and why the nearby gaps also play a role. Thus, given a large gap it would be useful to know what to expect for the nearby gaps. The motivation for understanding multiple eigenvalues comes from elliptic curved. The Birch and Swinnerton-Dyer conjecture (one of the Clay million-dollar problems) predicts that there are L-functions (generalizations of the zeta function) which have zeros of high multiplicity. There has been a small amount of theoretical work on how the multiple zero effects the nearby zeros, and I find some of those results counterintuitive. (I can give references).

What to do first: this project uses Haar measure on the unitary group U(N), as expressed in terms of the eigenvalues:
\begin{equation}
C_N \prod_{j<k} |e^{i\theta_j} - e^{i\theta_k}|\,d\theta_1\ldots d\theta_N .
\end{equation}
Given a set of eigenangles $\theta_1,\ldots,\theta_N$, the above expression tell how ``likely'' it is: more likely arrangements give larger numbers when plugged into the measure. Given two points on the circle with no other points between them, it is an optimization problem to find the locations of the other points which maximize the measure.

The project with multiple eigenvalues is similar, but you first have to figure out how to define the measure, because it is identically zero for multiple eigenvalues. I will add a reference for that.

Possible outcomes:This would be a good project for a student who wasn't necessarily planning to go to graduate school in math, because many businesses would be happy to have someone with experience in optimization methods. Also, if should be possible to make a nice movie showing points settling down to their maximum likelihood position.

Project 3:

Approach: I have a paper (with two REU students as coauthors) which will be posted on the ArXiv soon. The paper will give everything we know, and should be readable so you can decide if there is a good project there.

Project 4:

Background: Epstein zeta functions are generalization of the Riemann zeta function, except that (usually) they don't have an Euler product and (usually) do not satisfy a Riemann hypothesis. Nobody even has a conjecture for what proportion of zeros they have on the critical line. These function arise in number theory and in physics. Understanding these function should shed light on the zeta function.

What to do first: It is possible to generate these function on a computer, but it is much more difficult than the Riemann zeta function and it is hard to collect a lot of data. The current methods use know results about Eisenstein series -- See Hejhal's paper in the 1978(?) ICM proceedings. It should not be hard to implement that method, but it will be very slow.

Possible outcomes: faster way to compute, conjecture of proportion of zeros, which should help explain our inability to prove much about zeros of the zeta function.