Lillian
Pierce
Publications
Generalizations of the Schrodinger maximal operator: building arithmetic counterexamples
with R. Chu
(submitted)
(abstract)
This work studies methods for building counterexamples to pointwise convergence for dispersive PDE's with a real symbol belonging to a class of nonsingular homogeneous forms. In particular, by working with Dwork-regular forms and defining an appropriate notion of rank for such forms, we show that pointwise convergence of the solution to the relevant PDE must fail for some functions in a Sobolev space \(H^s\) in a range of \(s\) that goes above the previously known threshold 1/4. This advances previous understanding because our methods allow us to consider forms that are indecomposable (under any \(GL_n(\mathbb{Q})\) change of variable).
(hide abstract)
Generalized quadratic forms over totally real number fields
with T. D. Browning, D. Schindler
(submitted)
(abstract)
(arXiv)
We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy-Littlewood circle method over number fields.
(hide abstract)
A new type of superorthogonality
with P. T. Gressman, J. Roos, P.-L. Yung
(submitted)
(abstract)
(arXiv)
We provide a simple criterion on a family of functions that implies a square function estimate on \(L^p\) for every even integer \(p \geq 2\). This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known.
(hide abstract)
On polynomial Carleson operators along quadratic hypersurfaces
with T. C. Anderson, D. Maldague
(submitted)
(abstract)
(arXiv)
We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by a nondegenerate quadratic form \(Q\) (of any signature) is bounded a priori on \(L^p(\mathbb{R}^{n+1})\) for \( 1< p< \infty\) for all \( n \geq 2\). This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of a fixed set of homogeneous polynomials, where each such polynomial has degree at least 2, and the quadratic polynomial (if present) is not a multiple of the quadratic form \(Q\).
(hide abstract)
Application of a polynomial sieve: beyond separation of variables
with D. Bonolis
(submitted)
(abstract)
(arXiv)
This paper generalizes the polynomial sieve in order to count integral points \(\mathbf{x} \in [-B,B]^n\) for which an integral solution to \(F(y,\mathbf{x})=0\) exists. This is the first time a version of the polynomial sieve has been applied to count solutions in a case where \(y \) and \(\mathbf{x}\) are not separated, in the sense that \(F(y,\mathbf{x})\) has the special form \(f(y) = g(\mathbf{x})\).
(hide abstract)
Publications
Counting problems: class groups, primes, and number fields
Proceedings of the ICM 2022
(abstract)
(arXiv)
This paper surveys recent progress toward the \(\ell\)-torsion conjecture for class groups of number fields, and outlines its close connections to counting prime numbers, counting number fields of fixed discriminant, and counting number fields of bounded discriminant.
(hide abstract)
Geometric generalizations of the square sieve, with an application to cyclic covers
with A. Bucur, A. C. Cojocaru, and M. N. Lalin, with an appendix by J. Rabinoff
Mathematika 69 (2023) no. 1, 106-154.
(abstract)
(arXiv)
We formulate a general problem: given projective schemes Y and X over a global field
K and a K-morphism from Y to X of finite degree, how many points in X(K) of height at most
B have a pre-image in Y(K) under this morphism? This problem is inspired by a well-known conjecture of
Serre on quantitative upper bounds for the number of points of bounded height on an irreducible
projective variety defined over a number field. We give a non-trivial answer to the general problem
when K is \( \mathbb{F}_q(T)\) and Y is a prime degree cyclic cover of \( X = \mathbb{P}_K^n\). Our tool is a new geometric
sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.
(hide abstract)
On the strict majorant property in arbitrary dimensions
with P. Gressman, S. Guo, J. Roos and P.-L. Yung
Quarterly Journal of Mathematics (accepted)
(abstract)
(arXiv)
In this work we introduce the study of d-dimensional majorant properties.
This generalizes the notion of "majorant property" introduced by Hardy and Littlewood,
and known to have relations to Kakeya and restriction.
We prove that a set of frequencies in d dimensions satisfies the strict majorant
property on \(L^p\) for all \(p>0\) if and only if the set is affinely
independent. We further construct three types of violations of the strict
majorant property, both for generic sets of frequencies and for frequencies on the moment curve.
(hide abstract)
Counterexamples for high-degree generalizations of the Schrodinger maximal operator
with C. An and R. Chu
IMRN (available online)
(abstract)
(arXiv)
In 1980 Carleson posed a question on the minimal regularity of an
initial data function in a Sobolev space \(H^s(\mathbb{R}^n)\) that implies pointwise
convergence for the solution of the linear Schrodinger equation. After
progress by many authors, this was recently resolved (up to the endpoint)
by Bourgain, whose counterexample construction for the Schrodinger maximal
operator proved a necessary condition on the regularity, and Du and Zhang,
who proved a sufficient condition. Analogues of Carleson's question remain open
for many other dispersive PDE's. We develop a flexible new method to approach such
problems, and prove that for any integer \(k\geq2\), if a degree \(k\) generalization of the
Schrodinger maximal operator is bounded from \(H^s(\mathbb{R}^n)\) to \(L^1(B_n(0,1))\), then
\(s \geq \frac{1}{4}+\frac{n-1}{4((k-1)n+1)}.\)
In dimensions \(n \geq2\), for every degree \(k\geq3\), this is the first result that exceeds a
long-standing barrier at 1/4. Our methods are number-theoretic, and in particular
apply the Weil bound,
a consequence of the truth of the Riemann Hypothesis over finite fields.
(hide abstract)
On a conjecture for \(\ell\)-torsion in class groups of number fields: from the perspective of moments
with C. Turnage-Butterbaugh and M. M. Wood
Math. Res. Lett. 28 (2021) no. 2, 575-621.
(abstract)
(arXiv)
It is conjectured that within the class group of any number
field, for every integer \(\ell \geq 1\), the
\(\ell\)-torsion subgroup is very small
(in an appropriate sense, relative to the discriminant
of the field).
In nearly all settings, the full strength of this
conjecture remains open,
and even partial progress is limited. Significant recent
progress toward average
versions of the \(\ell\)-torsion conjecture has crucially
relied on counts for number fields,
raising interest in how these two types of question
relate. In this paper we make
explicit the quantitative relationships between the
\(\ell\)-torsion conjecture and
other well-known conjectures: the Cohen-Lenstra
heuristics, counts for number
fields of fixed discriminant, counts for number fields
of bounded discriminant
(or related invariants), and counts for elliptic curves
with fixed conductor.
All of these considerations reinforce that we expect the
\(\ell\)-torsion conjecture
is true, despite limited progress toward it. Our
perspective focuses
on the relation between pointwise bounds, averages, and
higher moments,
and demonstrates the broad utility of the method of moments.
(hide abstract)
On Superorthogonality
Journal of Geometric Analysis 31 (2021) no. 7, 7096-7183
with an appendix by Emmanuel Kowalski
(abstract)
(arXiv)
In this survey, we explore how superorthogonality amongst
functions in a sequence \(f_1, f_2, f_3,\ldots \)
results in direct or converse inequalities for an
associated square function. We distinguish between
three main types of superorthogonality, which we
demonstrate arise in a wide array of settings in
harmonic analysis and number theory. This perspective
gives clean proofs of central results, and unifies
topics including Khintchine's inequality, Walsh-Paley
series, discrete operators, decoupling,
counting solutions to systems of Diophantine equations,
multicorrelation of trace functions,
and the Burgess bound for short character sums.
(hide abstract)
Burgess bounds for short character sums evaluated at forms
II: the mixed case
Rivista di Matematica della Universita di Parma (N.S.) 12 (2021) no. 1, 151-179.
(abstract)
(arXiv)
This work proves a Burgess bound for short mixed character sums in \(n\) dimensions. The non-principal multiplicative character of prime conductor \(q\) may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed character sum is nontrivial when the length of the sum is at least \(q^{\beta}\) with \(\beta> 1/2 - 1/(2(n+1))\) in each coordinate. This work capitalizes on the recent stratification of multiplicative character sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimensions.
(hide abstract)
Reversing a philosophy: from counting to square functions
and decoupling
with P. Gressman, S. Guo, J. Roos, and P-Y Yung
Journal of Geometric Analysis 31 (2021) no. 7, 7075-7095.
(abstract)
(arXiv)
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality.
Second, in our main result we prove an \(L^{2n}\) square function estimate (which implies a corresponding decoupling estimate) for the extension operator associated to a non-degenerate curve in \(\mathbb{R}^n\). The proof is via a combinatorial argument that builds on the idea that if \(\gamma\) is a non-degenerate curve in \(\mathbb{R}^n\), then as long as \(x_1,\ldots, x_{2n}\) are chosen from a sufficiently well-separated set, then
\( \gamma(x_1)+\cdots+\gamma(x_n) = \gamma(x_{n+1}) + \cdots + \gamma(x_{2n}) \)
essentially only admits solutions in which \(x_1,\ldots,x_n\) is a permutation of \(x_{n+1},\ldots, x_{2n}\).
(hide abstract)
Elias M. Stein (1931-2018)
Notices of the AMS vol. 68 no. 4 April 2021, 546--563
with contributions by William Beckner, Galia Dafni, Charles Fefferman,
Alexandru Ionescu, Vickie Kearn,Carlos E. Kenig, Anthony W. Knapp,
Steven G. Krantz, Loredana Lanzani, Alexander Nagel, Duong H. Phong,
Fulvio Ricci, Linda Rothschild, Rami Shakarchi, Christopher Sogge,
Jeremy Stein, Karen Stein,Terence Tao, Stephen Wainger, and Harold Widom
(journal)
Analysis and Applications: The Mathematical Work of Elias Stein
Bulletin of the AMS (published online March 2020).
by C. Fefferman, A. Ionescu, T. Tao and S. Wainger; with contributions from L. Lanzani, A. Magyar, M. Mirek, A. Nagel, D. H. Phong, L. Pierce, F. Ricci, C. Sogge, B. Street.
(journal)
On Bourgain's counterexample for the Schrodinger maximal function
Quarterly Journal of Mathematics 71 (2020) no. 4, 1309-1344
(abstract)
(arXiv)
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrodinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.
(hide abstract)
An effective Chebotarev density theorem for families of
number fields,
with an application to \(\ell\)-torsion in class groups
with C. Turnage-Butterbaugh and M. M. Wood
Inventiones 219 (2) (2020) 701-778.
(abstract)
(arXiv)
In this work we prove a new effective Chebotarev density
theorem, independent of GRH, that improves the previously
known unconditional error term and enables the counting of
small primes; this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions.
The new effective Chebotarev theorem is expected to have many
applications, of which we demonstrate two. In particular,
we provide the first nontrivial upper bounds for
\(\ell\)-torsion, for all integers \( \ell \geq 1 \), applicable to infinite families of fields of arbitrarily large degree.
(hide abstract)
Burgess bounds for short character sums evaluated at forms
with J. Xu
Algebra and Number Theory 14 no. 7 (2020) 1911--1951.
(abstract)
(arXiv)
In this work we establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of admissible forms. This \(n\)-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least \(\gg q^{\beta}\), with \(\beta > 1/2 - 1/(2(n+1))\). This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the second author.
(hide abstract)
On matrix rearrangement inequalities
with R. Alaifari, X. Cheng, and S. Steinerberger
Proceedings of the AMS 148 (5) (2020) 1835-1848.
(abstract)
(arXiv)
Given two symmetric and positive semidefinite square matrices
\(A,B\), is it true that any matrix given as the product
of \(m\) copies of \(A\) and \(n\) copies of \(B\) in a
particular sequence must be dominated in the spectral norm
by the ordered matrix product \(A^m B^n\)? We prove that
for \(2\times 2 \) matrices, the general rearrangement
inequality holds for all disordered words. This contrasts
with work of Drury, which showed that this can fail for
larger matrices. Nevertheless, we also show
that for larger \( N \times N\) matrices, the general rearrangement inequality holds for all disordered words, for most \(A,B\) (in a sense of full measure) that are sufficiently small perturbations of the identity.
(hide abstract)
On
torsion subgroups in class groups of number fields,
JMM
2019 Sampler, Notices of the AMS, vol. 66 no. 1, January
(2019) 97-98
A polynomial Carleson operator along the paraboloid
with co-author Po-Lam Yung
Revista Mat. Ibero. 35 (2) (2019) 339-422.
(abstract)
(arXiv)
We consider a polynomial phase Carleson operator that is
of Radon type, namely it involves integration over a
paraboloid in Euclidean space of dimension at least 3. We
show that this operator is bounded on L^p for p strictly
between 1 and infinity, for certain classes of polynomial phase (in particular with
restrictions on the first and second order terms). 76 pages.
(hide abstract)
The Vinogradov Mean Value Theorem
[after Wooley, and
Bourgain, Demeter, Guth]
(Bourbaki Seminar, volume 69, 2016/2017, expose 1134)
Asterisque (2019) volume 407.
(abstract)
(arXiv)
(seminar)
This is the expository essay to accompany my June 2017 Bourbaki seminar
on the resolution of the Vinogradov Mean Value Theorem. 80 pages.
(hide abstract)
Endpoint Sobolev and BV Continuity for Maximal Operators
with co-authors E. Carneiro and J. Madrid
J. Functional Analysis, 273 (2017) 3262-3294.
(abstract)
(arXiv)
In this paper we investigate some questions related to the
continuity of maximal operators, complementing some
well-known boundedness results. We study the
one-dimensional uncentered Hardy-Littlewood maximal
operator in both the real-variable and discrete settings. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.
(hide abstract)
Polynomial Carleson operators along monomial curves in the plane
with co-authors Shaoming Guo, Joris Roos, Po-Lam Yung
Journal of Geometric Analysis (2017) 1-36.
(abstract)
(arXiv)
We prove L^p bounds for partial polynomial Carleson operators
along monomial curves in the plane with a phase
polynomial consisting of a single monomial. These
operators are "partial" in the sense that we consider
linearizing stopping-time functions that depend on only
one of the two ambient variables. A motivation for
studying these partial operators is the curious feature
that, despite their apparent limitations, for certain
combinations of curve and phase, L^2 bounds for partial
operators along curves imply the full strength of the L^2
bound for a one-dimensional Carleson operator, and for a
quadratic Carleson operator. 27 pages.
(hide abstract)
On \(\ell\)-torsion in class groups of number fields
with co-authors J. Ellenberg and M. Matchett Wood
Algebra and Number Theory 11 (8) Jan (2017) 1739-1778.
(abstract)
(arXiv)
For each integer n, we prove an unconditional upper bound
on the size of the n-torsion subgroup of the class group,
which holds for all but a zero-density set of field
extensions of the rationals of degree d, for any fixed d=2,3,4,5
(with the additional restriction in the case d=4 that the
field be non-D4). For sufficiently large n (specified
explicitly), these results are as strong as a previously
known bound that is conditional on GRH. As part of our
argument, we develop a probabilistic "Chebyshev sieve,"
and give uniform, power-saving error terms for the
asymptotics of quartic (non-D4) and quintic fields with
chosen splitting types at a finite set of primes. 25 pages.
(hide abstract)
Averages and moments associated to class numbers of
imaginary quadratic fields
with co-author D.R. Heath-Brown
Compositio. Math. 153 (2017) 2287-2309.
(abstract)
(arXiv)
For any odd prime q, consider the q-part of
the class number of the imaginary quadratic field
with square-free discriminant -d. Nontrivial pointwise upper
bounds are known only for q=3; nontrivial upper bounds
for averages of the q-part have previously been known
only for q=3,5 In this paper we prove nontrivial upper
bounds for the average of the q-part for all primes at
least 7, as well as nontrivial upper bounds for certain
higher moments for all primes at least 3. 25 pages.
(hide abstract)
Simultaneous integer values of pairs of quadratic forms
with co-author D.R. Heath-Brown
J. Reine Angew. Math. (Crelle) 727 (2017) 2287-2309.
(abstract)
(arXiv)
(journal)
We prove that a pair of integral quadratic forms in 5 or more
variables will simultaneously represent almost all pairs of integers
that satisfy the necessary local conditions, provided that the forms
satisfy a suitable nonsingularity condition. In particular such
forms simultaneously attain prime values if the obvious local
conditions hold. The proof uses the circle method, and
in particular pioneers a two-dimensional version of a Kloosterman
refinement.
(hide abstract)
Burgess bounds for multi-dimensional short mixed character sums
Journal of Number Theory, 163 (2016) 172-210.
(abstract)
(arXiv)
(journal)
This paper applies multi-dimensional Vinogradov mean value
theorems in order to prove Burgess bounds for
multi-dimensional short mixed character sums. 33 pages.
(hide
abstract)
Lower bounds for the truncated Hilbert transform
with co-authors R. Alaifari and S. Steinerberger
Revista Mat. Ibero., 32, no. 1 (2016) 23 - 56.
(abstract)
(arXiv)
(journal)
Given two intervals I and J on the real line,
we ask whether it is possible to reconstruct a
real-valued function supported on I from knowing its Hilbert transform
on the interval J.
When neither interval is fully contained in the other, this problem
has a unique answer (the nullspace is trivial) but is
severely ill-posed. We isolate the difficulty and show
that by restricting the problem to functions with controlled total variation, reconstruction becomes stable.
(hide abstract)
Representations of integers by systems of three quadratic forms
with co-authors D. Schindler and M. Matchett Wood
Proceedings of the London Mathematical Society (3) 113 (2016) 289-344.
(abstract)
(arXiv)
(journal)
We apply the circle method to count simultaneous
representations of "almost any"
tuple of three integers by a system of three quadratic
forms. In particular, we develop a three-dimensional
application of Kloosterman-style cancellation on the minor
arcs; inexplicit geometric criteria for smoothness play a
particularly important role. 66 pages.
(hide abstract)
Burgess bounds for short mixed character sums
with co-author D.R. Heath-Brown
Journal of the London Math. Soc., 91, no. 3 (2015) 693-708.
(abstract)
(arXiv)
(journal)
In this paper we introduce Vinogradov mean value theorems in
order to prove Burgess bounds for short mixed character sums.
(hide abstract)
Counting rational points on smooth cyclic covers
with co-author D.R. Heath-Brown
Journal of Number Theory, 132 (2012), pp. 1741-1757.
(abstract)
(pdf)
(arXiv)
(journal)
A conjecture of Serre concerns the number of rational
points of bounded height on a finite cover of (n-1)-dimensional projective
space. In this paper, we achieve Serre's conjecture in the
special case of smooth cyclic covers of any degree when n
is at least 10 and surpass it for covers of degree at
least 3 when n is at least 11. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method.
(hide abstract)
On a discrete version of Tanaka's theorem for maximal functions
with co-authors Jonathan Bober, Emanuel Carneiro, and Kevin Hughes
Proceedings of the Amer. Math. Soc. 140 (2012) 1669-1680.
(abstract)
(journal)
revised version on (arXiv)
In this paper we prove a discrete version of Tanaka's Theorem for the Hardy-Littlewood maximal operator in one dimension, both in the non-centered and centered cases. Up to date
version available on arXiv.
(hide abstract)
A note on discrete fractional integral operators on the Heisenberg group
Internat. Math. Res. Not. Vol. 2012, No. 1, (2012) 17-33.
(abstract)
(pdf)
(journal link)
We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.
(hide abstract)
Discrete fractional Radon transforms and quadratic forms
Duke Math. Journal 161, No. 1 (2012)
69-106. Correction 162 (2013) 1203-1204.
(abstract)
(pdf)
We consider discrete analogues of fractional Radon
transforms involving integration over paraboloids defined by
positive definite quadratic forms. We prove certain
conditions are sufficient for them to extend to bounded
operators from l^p to l^q for certain p,q. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.
(hide abstract)
On discrete fractional integral operators and mean values of Weyl sums
Bull. London Math. Soc., 43 (2011) 597-612.
(abstract)
(pdf)
(journal)
In this paper we prove new operator norms for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator.
From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number of representations of a positive integer as a sum of s positive k-th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.
(hide abstract)
A note on discrete twisted singular Radon transforms
Mathematical Research Letters 17, no. 4 (2010) 701-720.
(abstract)
(pdf)
(journal)
In this paper we consider three types of discrete
operators stemming from singular Radon transforms. We
first extend an l^p result for translation
invariant discrete singular Radon transforms to a class of
twisted operators including an additional oscillatory
component, via a simple method of descent
argument. Second, we note an l^2 bound for
quasi-translation invariant discrete twisted Radon
transforms. Finally, we extend an existing l^2 bound for a
closely related non-translation invariant discrete
oscillatory integral operator with singular kernel to an
l^p bound. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function.
(hide abstract)
A bound for the 3-part of class numbers of quadratic fields
by means of the square sieve
Forum Math. 18 (2006) no. 4, 677-698.
(abstract)
(pdf)
(journal)
This paper proves a nontrivial bound for the 3-part of the class number of a
real or imaginary quadratic field, by developing a variant of the square sieve and
the q-analogue of van der Corput's method.
(hide abstract)
The 3-part of class numbers of quadratic fields
J. London Math. Soc. (2) 71 (2005) 579-598.
(abstract)
(pdf)
(journal)
This paper proves non-trivial upper bounds for the 3-part
of the class number of real and imaginary
quadratic fields, as well as consequent upper bounds for the number of
elliptic curves with given conductor.
(hide abstract)
Oberwolfach report 2020 (link)
Oberwolfach report 2016 (link)
Oberwolfach report 2014 (link)
Oberwolfach report 2013 (link)
Princeton Lectures in Analysis, by Elias M. Stein and Rami Shakarchi — A Book Review
Book Review in the
Notices of the AMS, May 2012 issue
Reviewed by Charles Fefferman and Robert Fefferman, with contributions from Paul Hagelstein, Nataśa Pavlović, and Lillian Pierce
Discrete Analogues in Harmonic Analysis
PhD Thesis, Princeton University, 2009
Advisor: E. M. Stein
The
3-Part of Class Numbers of Quadratic Fields
MSc Thesis, Oxford University, 2004
Advisor: D. R. Heath-Brown
The
Pair Correlation of the Zeroes of the Riemann Zeta Function
Undergraduate Senior Thesis, Princeton University, 2002
Advisor: E. M. Stein
