title: Number Theory and Probability
Advisor: Steven J. Miller:
Click here for my schedule.
for different projects: Click here for pictures
Greetings. I have general ideas of the topics I want to pursue, but I often
don't finalize the projects until I know who is in the group and what their
interests are. Thus, you should view these as starting points of a conversation.
You can view all my papers from my
homepage. You'll find many papers from previous years of SMALL. The articles
below are meant to give you a rough sense of my interests, and once you tell me
what you find fascinating I'll provide more specific reading.
Description: We will explore many of the interplays between number theory
and probability, with projects drawn from L-functions, Random Matrix Theory,
Additive Number Theory (such as the 3x+1 Problem and Zeckendorf expansions) and Benfordís law. A common theme in many of these
systems is either a probabilistic model or heuristic. For example, Random Matrix
Theory was developed to study the energy levels of heavy nuclei. While it is
hard to analyze the behavior of a specific configuration, often it is easy to
calculate an average over all configurations, and then appeal to a Central Limit
Theorem type result to say that a generic systemís behavior is close to this
average. These techniques have been applied to many problems, ranging from the
behavior of L-functions to the structure of networks to city transportation. For
more on the connection between number theory and random matrix theory, see the
survey article by Firk-Miller.
Below is a reading list for the 2014 summer SMALL program in Number Theory
and Probability. It is important that we do some background reading and
have some ideas about what we are going to study before the program begins, as
we only have 9 weeks or so. It is thus essential to hit the ground running.
Obviously, I'm not expecting you to be able to read and thoroughly master
everything on the first pass. A lot of this reading is to give you the flavor of
the problems and the methods; many of these results we'll build on and thus it
is fine to accept them without fully understanding all the details of the
proofs. (Sadly, accepting some things on faith is necessary for short programs
in order to make progress; my hope is that you'll be interested enough to pursue
the material and learn the additional details.) There are more projects listed
than we can study; this is deliberate, so that we can choose projects based on
your interests and skill sets and not just on mine.
- General Reading
- Hayes: The
Spectrum of Riemannium: a light description of the connection between
random matrix theory and number theory (there are a few minor errors in the
presentation, basically to simplify the story). This is a quick read, and
gives some of the history.
- Conrey: L-functions and
Random Matrix Theory: This is a high level description of the
similarities between number theory and random matrix theory.
Zeros of Zeta Functions and Symmetry: Another high level article similar
to the others.
Patterns in Eigenvalues: this is a bit more readable than the others,
and is based on a distinguished lecture he delivered.
- Miller and Takloo-Bighash:
An Invitation to
Modern Number Theory: This is the textbook I and a colleague wrote,
based on years of supervising undergraduate research classes. I know several
of you already have a copy -- it will be a good resource for the summer, as
a lot of the background material we need is readily available here.
Particularly important chapters for us are:
(which discusses the connections between random matrix theory and number
theory, and is available online);
- chapter 18 (which does the explicit formula for Dirichlet
(which reviews L-functions and is also online).
- Firk and Miller:
Nuclei, primes and the Random Matrix connection: a survey paper on the
history of the subject, including both the nuclear physics experiments and
the theoretical calculations.
Low lying zeros of families of L-functions: This is a must read.
This is the first major paper calculating the 1-level density for families
of L-functions. We will not need much of the paper, but you must read
section 1, skim the beginning of section 2 (section 2 is devoted to
developing good averaging formulas for specific families; as we will be
looking at different families, we won't need this), skim section 3 (same
comments apply), section 4 (very important: here they prove the
explicit formula relating sums over zeros to sums over primes, and while we
will use the explicit formula for different families, the calculation will
follow similarly), section 5 (just skim, same caveats as before).
the L-functions Ratios Conjecture: This is a very recent conjecture
which is enjoying remarkable success in predicting answers. There is a lot
I'll say about this during the summer. I somewhat jokingly call it the
conjecture of the four lies, as there are five steps and four of the steps
are provably wrong (ie, the assumptions in those steps fail); however,
miraculously, all the errors seem to cancel to phenomenal level! I've become
very interested in testing this conjecture as much as possible, and have
written several papers in this (and have ideas for a few more which will be
very accessible to us). Read through page 17.
- Rubinstein: Low
lying zeros of L-functions and Random Matrix Theory: this is his
dissertation, and in it he analyzes the 1-level density of the family of
quadratic Dirichlet characters, and shows agreement with Random Matrix
Theory. This is one of the easiest families to look at, and a great testing
- Hughes - Miller: Low-lying
zeros of L-functions with orthogonal symmetry: this paper finds a more
tractable version of the Katz-Sarnak determinantal expansion for the n-level
density, but for restricted support.
symplectic test of the L-functions Ratios Conjecture: This paper builds
on those by Conrey-Snaith and Rubinstein and uses the Ratios Conjecture to
predict the lower order terms up to square-root cancellation, and then shows
(for suitable test functions) that this is the correct answer. An obvious
project is to generalize this test for other families or to enlarge the
- Miller: Extending the
support for families of Dirichlet characters (work in progress): This is
a project I've been working on occasionally over the past few years. I have
studied the 1-level density for Dirichlet characters, and subject to some
reasonable assumptions I can extend the support and smash all world records.
Is it possible to do this unconditionally? I don't know, but I hope so! I
would love to work on this during the summer; we will almost surely need the
results in the primes in arithmetic progression section below.
- Graph Theory
Eigenvalues of Large Random Graphs: This readable paper calculates the
density of eigenvalues for d-regular graphs; the answer is different than
the semi-circle Wigner found for the family of all real symmetric matrices.
Excellent projects are finding the density of states for special sets of
Random graphs: Numerics about the neighbor spacings of d-regular graphs.
- Womald: Models
of Random Graphs: various ways to generate random graphs.
- Graph Theory (Virus Propagation)
- Random Matrix Theory
- Benford's law of digit bias
- Additive number theory (MSTD sets)
- Additive number theory (Generalized
- Zeros of Random Polynomials
For additional projects, see
well as the project summary sheet at