My research is in the area of analytic number theory
.
Consider the problem of calculating $\pi(x)$ for a given positive number $x$, where $\pi(x)$ denotes the number of prime numbers not
exceeding $x$^{1}. While there exist exact formulas^{2} for $\pi(x)$, they
are computationally infeasible when $x$ is large. Therefore, it is desirable to approximate $\pi(x)$
using simpler quantities.

In 1896, Jacques Hadamard and Charles Jean de la Vallée Poussin independently proved the prime number theorem , which roughly states that $\frac{x}{\log(x)}$ is a good approximation of $\pi(x)$ for large values of $x$, where $\log$ denotes the natural logarithm. Their proof depended on properties of the Riemann zeta function , which is denoted by $\zeta$. Obtaining deeper understanding of $\zeta$ in the critical strip is key to improving our knowledge of the sequence of prime numbers. For example, the Riemann hypothesis , which is one of the most important unsolved problems in mathematics, implies a significant improvement of the prime number theorem:

\[|\pi(x) - \text{li}(x)| \leq C \sqrt{x}\log(x),\]

where $x$ is large, $\text{li}$ denotes the logarithmic integral function , and $C$ is some fixed number independent of $x$. It is important to note that $\text{li}(x)$ behaves like $\frac{x}{\log(x)}$ for large values of $x$.

Another important unsolved problem about the behavior of $\zeta$ is the
Lindelöf hypothesis
, which claims a strong
bound on the size of $\zeta$ on the critical line
.
My current research deals with investigating such bounds for families of
automorphic $L$-functions
,
which are a vast generalization of $\zeta$. Specifically, I have investigated $q$-aspect subconvexity
bounds for L-functions on $GL(3)\times GL(2)$ and $GL(3)$ using the moment approach. For more details
about my research, please feel free to take a look at these **talk slides
**.

I am also interested in applications of algebraic geometry to analytic number theory, especially
to the bounding of character sums^{3}. Over the course of my undergraduate and master's education,
I have received formal training in theoretical computer science, statistics, probability, and computer
programming, and I often find myself applying knowledge from these areas to problems in number theory.

^{1}

Not to be confused with the constant $\pi$ times $x$. Please see prime-counting function for more details.

^{2}

^{3}

This lecture series by Fouvry, Kowalski, Michel, and Sawin is a great reference for such techniques.