Research Papers

Click on title for abstract

  1. Pair correlation of $L$-zeros and Hooley's conjecture. Available upon request.

    Abstract. In this article, we study the form factor for zeros of Dirichlet $L$-functions, pairing zeros within each primitive character and then averaging over primitive characters modulo a prime conductor. Unconditionally, we show for $\vert\alpha\vert<1$ that the form factor agrees with the random matrix prediction from the Circular Unitary Ensemble (CUE). Under GRH and the Hardy–Littlewood twin primes conjecture, we extend this agreement to $\vert\alpha\vert<2$. For $\vert\alpha\vert\ge1$, we prove that the CUE prediction is equivalent to Hooley's conjecture for the variance of primes in arithmetic progressions with prime moduli. We also explore applications of these form-factor results to the simplicity of zeros.

  2. Explicit quadratic large sieve inequality. Acta Arithmetica, 223(3):227-252 (2026).

    Abstract. We obtain an explicit version of Heath-Brown’s large sieve inequality for quadratic characters and discuss its applications to $L$-functions and quadratic fields.

  3. On sums of Fourier coefficients of cusp forms twisted with additive characters (2023). arXiv.

    Abstract. When $a_n$ is the $n$'th coefficient of some holomorphic cusp form, we prove a variety of Omega results for the twisted sum $\sum_{n\le x}a_ne^{2\pi i n\alpha}$ and discuss their applications to the Ramanujan $\tau$-function and sums of $a_n$ over arithmetic progressions.

  4. On a weighted sum over multiplicative functions and its applications to the GPY sieve (2022). arXiv.

    Abstract. In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's smoothened GPY sieve with coefficient expressions that are friendly to numerical computations.

  5. A corrected simplified proof of Chen’s theorem (2022). arXiv.

    Abstract. In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simplified proof of this result given by Pan, Wang, and Ding.

Expository

Random matrix theory and $L$-functions

Additive divisor problem

Gaps between primes

Lectures on sieve methods (Chinese version)

Talks

  • 10 Aug 2026: Random matrix theory and $L$-functions (Slides) (Recording)

  • 21 July 2025: On the distribution of zeros of $\zeta(s)$ (Slides) (Recording)

  • 9 Dec 2024 (with Ruiyang Tang and Siyu Liu): Inverse Galois problem (Slides)

  • 16 Oct 2024: Special values of the Riemann zeta function (Slides) (Recording EP1 EP2)

  • 28 Feb 2024 (with May Jiang, Qing Su, and Hantang Guo): Introduction to $p$-adic Analysis (Slides) (Recording)

  • 11 Oct 2023: Ray Reflection and Rational Approximation (Notes) (Recording)

  • 6 June 2023 (with Yi Liu, Daya Singh, and Tairan Wang): Axiom of Choice: Equivalents, Consequences, and Independence (Slides)

  • 10 Mar 2023: Ellipses, Pendulum, and Double Periodicity (Notes) (Recording)

  • 28 Oct 2022: Bounded gaps between primes (Slides) (Recording)

The recordings are also available on Bilibili.