Research

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 formulas² 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 sums³. 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.