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. 32 + 12), 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 23 * 32 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
