# Research

My branch of expertise is analytic number theory, which means that I’m interested in studying properties of the integers through analytic means.  This field is typically divided into additive number theory (which studies sumsets and open problems like the Goldbach Conjecture and Waring’s Problem using the Hardy-Littlewood Circle Method) and multiplicative number theory (which studies the properties of L-functions as a way to understand things like the distribution of primes).

My research interests lie in multiplicative number theory and the analytic theory of L-functions, especially as regards application of that theory to classic and/or elementary problems in number theory, including the

Brief descriptions of my published work can be found below:

[-] Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences
(Joint with Z. K. Li; view on arXiv, ResearchGate)

A sequence $\{a_n\}$ is holonomic over $\mathbb{Z}$ if its associated power series $\sum a_n t^n$ satisfies a differential equation over $\mathbb{Z}[t]$.  Holonomic sequences appear in a certain Gauss–Manin connection called the Picard–Fuchs equation.  Here, we establish criteria to determine when holonomic sequences are not just rational, but integral (in which case, there is typically a geometric explanation).  Some asymptotics for the growth of holonomic sequences are also given.

[-] The Second Moment of Sums of Coefficients of Cusp Forms
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

This paper begins a multi-paper investigation into various properties of sums of coefficients of holomorphic cusp forms.  Bounds for these sums can be thought of as abstracted versions of the circle and divisor problems, since each of these problems consider partial sums of coefficients of modular functions.  In this paper, we construct a Dirichlet series whose coefficients are the squares of these partial sums, provide a full meromorphic continuation for this series, and use this to prove a smooth mean-square estimate (with main term and error) for sums of Fourier coefficients of cusp forms.

[-] Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

A `classical conjecture’ states that the coefficients of a weight-holomorphic cusp form $f(z) = \sum a(n) e(nz)$ should satisfy

$\displaystyle S_f(X) := \sum_{n \leq X} a(n) \ll X^{\frac{k-1}{2}+\frac{1}{4}+\epsilon}.$

In this paper, we use the analytic properties of series introduced in (1) to prove that this conjecture holds in mean-square over short intervals of the form $[X, X + X^{\frac{2}{3}+\epsilon}]$.  This improves work of Jutila, who proved an analogous result with exponent $3/4$.

[-] Sign Changes of Coefficients and Sums of Coefficients of L-Functions
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

Meher-Murty give an axiomatization for detecting sign changes of coefficient series in short intervals of the form $[X, X+X^\alpha]$.  In this paper, we generalize this axiomatization in various ways, eg. when the coefficient series represents the Fourier coefficients of an  L-function.  Building upon prior work, we show that the sums of coefficients of cusp forms experience sign changes in short intervals.  (Note that sign changes for partial sums imply sign changes for individual coefficients.)

[-] Graphs of Large Girth and Surfaces of Large Systole
(Joint with Bram Petri; view on arXiv, ResearchGate)

In this paper, we consider the problem of construction graphs with no short loops (ie. graphs of large girth) and surfaces with no short (homotopically non-trivial) loops (ie. surfaces of large systole).  Our construction of the former makes use of an arithmetic lemma on the number of $\mathrm{SL}_2(\mathbb{Z})$ matrices of small trace.  This paper is similar in scope to work of Lubotzky, Philips, and Sarnak, who showed that the arithmetic properties of a different class of graphs (Ramanujan graphs) can produce similar results.  (More information about this paper can be found in this post.)

[-] Second Moments in the Generalized Gauss Circle Problem
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

The generalized Gauss Circle Problem is the problem of estimating the number of lattice points within a k-dimensional sphere of large radius.  Typically, one applies the circle method to produce such estimates.  Here, we show that the spectral theory of automorphic forms can also produce good results.  This paper produces the first power-savings second moment result in the 3-dimensional Gauss circle problem.

[-] A Shifted Sum for the Congruent Number Problem
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

The congruent number problem is the classification problem of determining which integers appear as the areas of right triangles with rational side lengths.  This paper studies a connection between the congruent number problem and the asymptotics (main term and error) of a quadruple shifted convolution sum.

[-] Arithmetic Progressions with Restricted Digits
(Joint with Aled Walker; view on arXiv, ResearchGate)

This paper determines the exact maximal length of an arithmetic progression which, when written in base-b, omits at least one digit.  We show that this maximal length is asymptotically $b^2$ outside of a density 0 set.

[-] The Laplace Transform of the Second Moment in the Gauss Circle Problem
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

This paper, a complement to Second Moments in the Generalized Gauss Circle Problem, applies the spectral theory of automorphic forms to produce second moment results in the classic (planar) Gauss circle problem.  Sharp cutoff results are unremarkable, but smooth analogues (i.e. the Laplace transform) improve on known results and are essentially optimal.

[-] Triple Correlation Sums of Coefficients of Cusp Forms
(Joint with Thomas A. Hulse, Chan Ieong Kuan, and David Lowry-Duda; view on arXiv, ResearchGate)

This paper produces estimates for the growth of triple correlation sums of the form

$\displaystyle \sum_{m \leq X} \sum_{h \leq Y} a(h)b(m)c(2m-h),$

in which $a(\cdot)$, $b(\cdot)$, and $c(\cdot)$ are the coefficients of holomorphic cusp forms.  We demonstrate square-root type cancellation in sums on and 3/4-power cancellation in sums on h, the latter conditional on the Riemann hypothesis.  Our results imply that the coefficients of cusp forms are simultaneously non-vanishing on infinitely many arithmetic progressions.