Papers and Talks on Random Matrix
Theory and L-functions
Below is a reading list and slides / notes on talks for the
Workshop on Zeta Functions, L-Functions and their Applications. Obviously
the links below are just a subset of the vast literature. Some articles I
haven't received permission to post yet, so email me at sjm1 AT williams.edu and
I'll send you a copy. NOTE: This webpage is very much a work in progress! If
you know of other papers that would be relevant, please send me a paper and I'll
add it below.
HERE FOR THE CONFERENCE NOTES
First is some general reading, then links to papers by Conrey and his
colleagues (that will be the basis of his talks) and slides for Miller's talk.
- General Reading
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 (email me for a copy).
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 characters)
(see me if you want a copy).
(which reviews L-functions and is also online).
principle L-functions and random matrix theory: This paper analyzes the
n-level correlations of zeros of automorphic L-functions and shows agreement
with random matrix theory; included in the paper are bounds towards
Ramanujan and the explicit formula for GL(n). For more papers, see
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
- Hughes-Miller: Low lying
zeros of L-functions with orthogonal symmetry. This paper generalizes
the results of Iwaniec-Luo-Sarnak to the n-level density. The difficulty is
in handling the combinatorics to show agreement with RMT.
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 ground.
published paper (in Duke) is here.
- Conrey-Snaith: Applications
of the L-functions Ratios Conjecture: This is a very recent conjecture
which is enjoying remarkable success in predicting answers. 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).
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
- Duenez-Miller: The effect of
convolving families of L-functions on the underlying symmetry. In
this paper we show how one may determine the corresponding classical compact
group for convolutions of certain families of L-functions. This paper was
motivated by The low-lying zeros
of a GL(4) and a GL(6) family of L-functions (Duenez-Miller), which
disproved a folklore conjecture on the corresponding classical compact
- Miller (with an appendix by Duenez):
Investigations of zeros near the
central point of elliptic curve L-functions. In this paper we look at
the experimental data of the first few zeros above the central point in
families of elliptic curves, with and without rank. We see the effect of
rank, and are led to certain conjectures as to the behavior of low-lying
zeros for finite conductors. We know what the behavior of these zeros is in
the limit as the conductors tend to infinity; see
1 and 2 level densities for
rational families of elliptic curves (Steven J Miller) and
Low-lying zeros of families of
elliptic curves (Matthew Young).
- Computational Papers
Below are links to
papers by Brian Conrey (which will be the basis of his talk):
Notes on eigenvalue distributions for the classical compact groups
- Sections of the paper: Haar measure, Vandermonde determinants and
orthogonal polynomials, Andreief's identity, Gaudin's lemma, n-level density,
correlations, neighbor spacings.
Integral Moments of L-functions (with Farmer, Keating, Rubinstein and
- Abstract: We give a new heuristic for all of the main terms in the
integral moments of various families of primitive L-functions. The results
agree with previous conjectures for the leading order terms. Our conjectures
also have an almost identical form to exact expressions for the corresponding
moments of the characteristic polynomials of either unitary, orthogonal, or
symplectic matrices, where the moments are denoted by the appropriate group
averages. This lends support to the idea that arithmetical L-functions have a
spectral interpretation, and that their value distributions can be modeled
using Random Matrix Theory. Numerical examples showgood agreement with our
of the L-functions Ratios Conjecture (with Snaith)
- Abstract: In upcoming papers by Conrey, Farmer and Zirnbauer there appear
conjectural formulas for averages, over a family, of ratios of products of
shifted L-functions. In this paper we will present various applications of
these ratios conjectures to a wide variety of problems that are of interest in
number theory, such as lower order terms in the zero
statistics of L-functions, mollified moments of L-functions and discrete
averages over zeros of the Riemann zeta function. In particular, using the
ratios conjectures we easily derive the answers to a number of notoriously
On the frequency of vanishing of quadratic twists of modular L-functions
(with Keating, Rubinstein and Snaith)
- Abstract: We present theoretical and numerical evidence for a random
matrix theoretical approach to a conjecture about vanishings of quadratic
twists of certain L-functions.
- Information on his talk:
Below are links to slides for some talks at
- Miller: From Random Matrix Theory to
L-functions. Summary: I explain classical RMT and show the three key
features in common b/w RMT and Number Theory: (1) find the correct scale to
study events; (2) develop an explicit formula to related what you want to
study to what you can study; (3) develop averaging formulas to analyze the
quantities we can study. For example, in RMT we have the Eigenvalue Trace
Lemma as our explicit formula, which combined with the CLT gives the correct
scale and then simple polynomial integration and combinatorics give the
averaging formulas. The next section reviews some properties of zeta(s) and
L-fns and states the various statistics one studies (correlations, level
densities). The ext section analyzes in great detail the 1-level density for
families of Dirichlet characters. For prime conductors this was done by Hughes
and Rudnick; I independently did it as a warm-up to my thesis but in greater
generality (square-free modulus). Almost surely there is too much info here,
but if someone wants the details, it's here. The reason I like this example is
the averaging formulas are simpler than other families of L-fns, and thus one
can get a sense of the theory without getting caught up in too much technical
detail. The last section is the technical detail section, trying to show how
one proves the 1 or n-level density results for cuspidal newforms (ie, the
Iwaniec-Luo-Sarnak paper and my work with Chris Hughes). This is very
technical, but I try to show what terms appear and how we handle them
(obviously only in the most general terms).