Research

My research in analytic number theory 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.

Besides problems of analytic nature, I am also interested in studying sets of integers satisfying certain additive constraints. My research with Andrew Lott on sets whose difference sets avoid certain patterns, such as squares, sums of two squares, and shifted primes, has contributed to the advancement of our understanding of how dense such sets can be. Recently, we obtained the optimal density bound for sets whose difference sets avoid perfect squares that one can hope to achieve using the quantitative van der Corput properties, improving upon the previous best bound due to Green and Sawhney. In a different direction, my recent work on strongly complete sets showed that if a set of positive integers has enough elements locally and satisfies some divergence condition, then every sufficiently large integer can be expressed as a sum of distinct elements from this set. In particular, this resolves a 1961 conjecture of Erdős.