Research

My research centers on how prime numbers distribute among natural numbers and interesting applications of such information to various problems in analytic and probabilistic number theory. I am particularly interested in the distribution of primes in intervals and arithmetic progressions as well as in gaps between consecutive primes. Well-known unsolved problems in these directions include the Riemann Hypothesis and the Twin Prime Conjecture. To study primes, I use a combination of tools from complex analysis, harmonic analysis, sieve theory, probability theory and combinatorics, and this is where my area of research interacts with other areas in exciting ways.

Studying the distribution of primes is important in that primes are the building blocks of natural numbers and information about them is therefore indispensable in the study of plenty of arithmetic objects and structures. One such example is arithmetic functions, which are functions defined on natural numbers. In my research, I study the mean values of these functions as well as their distributions when modeled as appropriate random variables. As in my study of primes, I apply a variety of techniques to handle problems about arithmetic functions. Not only does my research find its applications in my field, but it also sheds light on some objects from algebraic number theory. For instance, my research on weighted Erdős–Kac theorems leads to a new proof of a central limit theorem for Ramanujan’s τ-function due to Elliott, and my recent work with Paul Pollack on an analogue of Carmichael’s function plays an important role in the study of the typical size of an order in a fixed quadratic number field.