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:

[1] The Second Moment of Sums of Coefficients of Cusp Forms (Joint work with Thomas Hulse, Chan Ieong Kuan, and David Lowry-Duda)

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.

[2] Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms (Joint work with Thomas Hulse, Chan Ieong Kuan, and David Lowry-Duda)

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 $3/4$.

[3] Sign Changes of Coefficients and Sums of Coefficients of L-Functions (Joint work with Thomas Hulse, Chan Ieong Kuan, and David Lowry-Duda)

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.)

[4] Graphs of Large Girth and Surfaces of Large Systole (Joint work with Bram Petri)

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.)

[5] Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences (Joint work with Zane Kun Li)