Doctoral Research
My research work during my Ph.D. has been in the field of analytic number theory, primarily focused on arithmetic functions and partial sums of their associated Dirichlet series. Under the assumption of the Riemann hypothesis, there are well-known estimates for these partial sums. I have been working on investigating and establishing equivalences between the Riemann hypothesis and estimates on partial sums of various twisted Dirichlet series. I worked with my advisor Dr. William Banks to establish such an equivalence for k-convolutions of the von Mangoldt function and the generalized von Mangoldt functions.
Here's a link to the preprint.
Currently, I am investigating the connection between estimates on partial sums of other arithmetic functions and the Riemann hypothesis. In particular, the arithmetic functions of interest to me are the ones whose Dirichlet series can be expressed in terms of the Riemann zeta function, such as the Louiville lambda function or the Euler-totient function.
Master's Research (2017-2018)
My master's research was on the topic of rational points on elliptic curves under the guidance of Dr. Narasimha Kumar In this two-semester project, I studied the arithmetic structure of rational points on elliptic curves, including the classification of elliptic curves up to isomorphism. I also explored various different families of elliptic curves such as the Legendre and Hessian families of elliptic curves, and studied the proof of one of the major theorems in the theory of elliptic curves - the Mordell-Weil finite generation theorem.
Undergraduate Summer Research (2015)
I was selected to do undergraduate research at IISER (Indian Institute of Science Education and Research) Mohali after completing my second year of undergrad. I worked under the guidance of Dr. Amit Kulshrestha. For my summer research, I studied the group structure of rotations of Platonic solids. I worked on a combinatorial counting problem regarding different ways to color the faces, and vertices of edges of Platonic solids using n different colors. I solved this problem with the application of group theory and Burnside's lemma.