# Research

I am interested in different number theoretic questions.

During my graduate school and shortly after, I worked on the **pair correlation conjecture** and its connection to distribution of prime numbers. Below are some samples.

More precise pair correlation of zeros and primes in short intervals

Pair Correlation of the zeros of the Riemann zeta function in longer ranges

A note on Primes in Short Intervals

My Ph.D. advisor, Prof. Hugh L. Montgomery, also got me interested in special integer
sequences in **short intervals** such as perfect powers, squarefree numbers, ... . Related to perfect squares and
sums of two squares (e.g. 3^{2} + 1^{2}), I became interested in almost squares (e.g. 9 * 11), numbers with two close factors.

On moments of gaps between consecutive squarefree numbers

I am also interested in **diophantine approximation** problems with a surprising link to Ramanujan sums.

Approximating reals by sums of two rationals

Approximating reals by sums of rationals

Studying short-interval and rational approximation problems led me to study close
**lattice points** on curves over rationals and finite fields. It has connection to Kloosterman sums,
character sums, and least quadratic nonresidues.

An almost all result on q1q2≡c(modq)

Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue

Shortest Distance in Modular Cubic Polynomials

Closely related to almost squares is the topic of **close divisors** of an integer and lattices points close to a curve. I got into this area after reading
a paper by Erdos and Rosenfeld. It also led me to study gaps on "divisibility sequences".

Perfect squares have at most five divisors close to its square root

Factors of almost squares and lattice points on circles

Numbers with three close factorizations

Gaps between divisible terms in a2(a2+1)

Speaking of "divisibility sequences", it was partially motivated by Erdos' conjecture
on **primitive sequences** (which was recently resolved by Jared D. Lichtman in 2023). Before that, we worked
together on some closely related questions.

A generalization of primitive sets and a conjecture of Erdős

On the critical exponent for k-primitive sets

Throughout my career, I was and am still fascinated with **squarefull / powerful numbers** like 2^{3} * 3^{2} with exponents greater than 1 in its prime factorization. They are somewhat like
perfect squares but with more irregularities. I am interested in their behavior in
short intervals, along arithmetic progressions as well as over multiplicative functions.
A prime motivation is Erdo's conjecture: There is no three consecutive squarefull
numbers.

Squarefull numbers in arithmetic progression II

Variance of Squarefull Numbers in Short Intervals

Variance of squarefull numbers in short intervals II

A note on powerful numbers in short intervals

Arithmetic progressions among powerful numbers