Research

Papers:

The Sign of Fourier Coefficients of Half-Integral Weight Cusp Forms,  joint with E. Mehmet Kıral, Chan Ieong Kuan 
and Li-Mei Lim. 
International Journal of Number Theory, Vol. 8, No. 3 (2012) 749-762

Multiple Dirichlet Series and Shifted Convolutions, joint with Jeffrey Hoffstein.
Journal of Number Theory, Volume 161, April 2016, Pages 457-533



Counting Square Discriminants, joint with E. Mehmet Kıral, Chan Ieong Kuan, and Li-Mei Lim.
Journal of Number Theory, Volume 162, May 2016, Pages 255-274

Bertrand's Postulate for Number Fields, joint with M. Ram Murty.
In Press. Colloquium Mathematicum

The Second Moment of Sums of Coefficients of Cusp Forms, joint with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker.
Journal of Number Theory, Volume 173, April 2017, Pages 304-331

Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms, joint with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker.
Journal of Number Theory, Volume 173, April 2017, Pages 394-415

Sign Changes of Coefficients and Sums of Coefficients of L-Functions, joint with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker.
Accepted, Journal of Number Theory

Triple Shifted Sums of Automorphic L-functions (Ph.D Thesis)

In Progress:

The Sign of Fourier Coefficients of Half-Integral Weight Cusp Forms II.
joint with Jianing Yang.

Lattice Points in Spheres
joint with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker.

Effective Sign-Change Intervals for Hilbert Modular Forms
joint with Naomi Tanabe.

Selected Talks:

-October 2016 Lattice Points in Spheres Motivated by Gauss's Circle Problem and Dirichlet's Divisor Problem, a talk on the analogous question of finding asymptotics for the partial sums of Fourier coefficients of automorphic forms. More specifically, an introduction to how this problem can be approached via shifted multiple Dirichlet series and our results pursuing this. This is joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker.

-January 2016 Generalizing Sign-Change Axiomatizations and Average Orders
Here we present new results about the asymptotic behavior of average orders of the Fourier coefficients of holomorphic cusp forms by means of meromorphically continuing Dirichlet Series whose coefficients are squares of these partial sums. We do this by decomposing these Dirichlet series into shifted Multiple Dirichlet Series and taking spectral expansions. More recently, we have begun to generalize this construction to non-cusp forms to investigate the generalization of Gauss circle problem to higher dimensions. This is joint work with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker.

-November 2015 Finding Imaginary Primes, Colby College Departmental Colloquium
An introduction to primality, Bertrand's postulate, the prime number theorem, and how these all can be extended to arbitrary number fields with the Gaussian integers as a concrete example. A new result, stating a generalization of Bertrand's postulate for number fields, is given. Intended for a mixed audience of undergraduates and faculty. This is joint work with M. Ram Murty.

-October 2015, Averaging Average Orders, Maine-Québec Number Theory Conference
Motivated by Gauss's Circle Problem and Dirichlet's Divisor Problem, a talk on the analogous question of finding asymptotics for the partial sums of Fourier coefficients of automorphic forms. More specifically, an introduction to how this problem can be approached via shifted multiple Dirichlet series and our results pursuing this. This is joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker.

-October 2014, Counting Square Discriminants, Queen's University Departmental Colloquium
I presented the problem of counting indefinite binary quadratic forms with fixed discriminant given certain restrictions on the coefficients, in particular I outlined how to obtain a novel result when the discriminant is a square. I showed that this information could be obtained by investigating the analytic properties of a shifted multiple Dirichlet series comprised of well-understood arithmetic functions and that this analytic information is derived by taking a spectral expansion of a particular truncated Poincaré series inspired by a similar creation due to Jeffrey Hoffstein. This is joint work with E. Mehmet Kıral, Chan Ieong Kuan and Li-Mei Lim.

-June 2014, The Sign of Fourier Coefficients of More General Half-Integral Weight Cusp Forms, CNTA XIII
In a 2012 paper written with E.M. Kıral, C.I. Kuan and L. Lim, we showed that the square-free coefficients of a half-integral weight cusp form change sign infinitely often, given additional restrictions that this cusp form was a level four Hecke eigenform with real coefficients. More recently, I have been working to loosen those additional restrictions, motivated by a 2013 sign-changing axiomatization theorem due to J. Meher and M. Ram Murty. This updated argument hinges on a better understanding of the Rankin-Selberg L-function of two half-integral weight Hecke eigenforms twisted by an additive character and its functional equation. In this talk I outlined the modified argument and the progress made thus far.

-December 2013, Some Results of Shifted Sums, Invited Talk, Number Theory Session, CMS Winter Meeting
Knowledge about shifted convolution sums can be used to bound different aspects of automorphic L-functions and thus make piecemeal progress toward the General Lindelöf Hypothesis. In joint work with Jeffrey Hoffstein, joint work with E. Mehmet Kıral, Chan Ieong Kuan and Li-Mei Lim, and my own thesis work, we investigated the application of a different sort of truncated Poincaré series, proposed by Hoffstein, in our study of particular shifted sums. These sums have in turn been used to obtain non-trivial asymptotics of triple sums of Fourier coefficients of classical holomorphic cusp forms, a Burgess-type bound for twisted automorphic L-functions, and a smoothed count of square discriminants with bounded coefficients.

-October 2012, A Double Spectral Expansion of a Triple Shifted Dirichlet Series, Brown Algebra Seminar
An introduction to shifted convolutions are and how knowledge about them can be used to give subconvexity bounds of automorphic L-functions. I discuss my own work, based off of the work of Jeff Hoffstein, which uses shifted convolutions of holomorphic forms with Maass forms to give a continuation of a triple, shifted Dirichlet series.


Pictured above, from left to right: me, E. Mehmet Kıral and Chan Ieong Kuan