Research

I am interested in analytic number theory and algebraic number theory. The main focus of my research is the effective Chebotarev density theorem and several related problems, including l-torsion in class groups and zero density estimates of L-functions.

The Chebotarev density theorem is one of the most important theorems in number theory. Its beauty and importance are clearly shown when we consider the Prime Number Theorem for arithmetic progressions, which is a particular case of the Chebotarev density theorem. The Prime Number Theorem for arithmetic progressions is a result of counting primes and states the density of the primes with a congruence condition among all primes. The error terms of both theorems are closely connected with the Generalized Riemann Hypothesis, one of the most famous open problems in mathematics.

For the first project of my research, I worked on improving the effective Chebotarev density theorem and on its application to l-torsion of class groups. My paper l-torsion in class groups of certain families of D4-quartic fields gave the first unconditional results of effective Chebotarev density theorem with a small threshold and of l-torsion of class groups, for certain families of D4-quartic fields. It is vital to have the small threshold for its own sake and for the advantage of counting small split primes, which leads to a better bound of l-torsion of class groups. The difficulty of the families of D4-quartic fields is that these families cannot be treated by the previous methods due to the special group structure of D4.

For my current research, I am working on zero density estimates of automorphic L-functions. A zero density estimate counts the number of possible zeros of L-functions in a specific region; hence, it may help us understand the Riemann Hypothesis, arguably the most famous conjecture in mathematics. The zero density estimate has applications to the Chebotarev density theorem and the l-torsion in class groups. It also has applications to Linnik’s theorem. My current result improves the previously best-known zero density estimate for automorphic L- functions.

For future research, I am trying to obtain new Chebotarev density theorems, based on my original zero density estimate. The new Chebotarev density theorems are expected to hold for any general base field with a small threshold such that the theorems are valid.