数论讨论班

摘要：In this talk, we investigate the value distribution of Hecke eigenforms in the large weight limit. We begin with an introduction to the quantum unique ergodicity (QUE) theorem and its application to the equidistribution of zeros of Hecke eigenforms. We then focus on their joint value distribution. In particular, we establish asymptotic formulas for certain low-degree mixed moments. Our approach is based on the analytic theory of L-functions.

摘要：In this talk, I'll show that there are secondary terms in the mean values of Maass special L-values. This curious phenomenon is rare in the moment conjectures and is known only for the cubic moment of quadratic Dirichlet central L-values.

摘要：For a nonempty set A of integers and an integer n, let rA(n) be the number of representations of n of the form n = a + a′, where a ≤a′ and a, a′ ∈A, and dA(n) be the number of (a, a′) with a, a′ ∈A such that n = a −a′. In this talk, we will present a review of our results on representation functions.

摘要：我们将讨论关于素数小间隔的Zhang-Maynard-Tao 定理的一个Khinchin 型的推广。

摘要：TBA

摘要：In this talk, I will explain a relative version of Brunn-Minkowski inequality and its consequence in Arakelov geometry in the form of an inequality of arithmetic intersection numbers.

摘要：In this talk I will first introduce some basic knowledge about relative trace formulae. Then I will talk about two applications of it to analytic number theory: the cubic moment problem for the L-functions of automorphic representations of GL(2), and subconvexity problem in level aspect for L-functions on GL(n).

摘要：This is joint with Brian Conrey, David Farmer, Chung-Hang Kwan, and Caroline Turnage-Butterbaugh. When studying the zeros of Riemann zeta function at a height T up the critical strip one often multiplies zeta by a Dirichlet polynomial, called a mollifier, of length Tθ before averaging in order to neutralize the irregularities of zeta. Levinson in his 1974 Advances paper famously proved that at least 1/3 of the zeros of zeta are on the critical line, by using a mollifier of length Tθ with θ < 1/2. Significant efforts in the literature have been devoted to refine and optimize Levinson's mollifer. We prove that Levinson's method, as modified by Conrey, will in fact produce a positive proportion of critical zeros, regardless how short the mollifier is.

摘要：The prime geodesic theorems (PGT) are analogues of the prime number theorem (PNT), in which we count the primitive closed geodesics in Γ\ℍ instead of prime numbers. Here $Γ < PSL_2(ℝ)$ is a lattice, and ℍ is the Poincaré upper half plane. In the setting of co-compact lattices constructed from quaternion algebras, the record is kept by Koyama for full level subgroups at Luo-Sarnak's level 7/10. Koyama's method is an exploitation of the Jacquet-Langlands correspondences on the spectral side.

We generalize Koyama's 7/10 bound of the error term in the prime geodesic theorems to the principal congruence subgroups for quaternion algebras. Our method avoids the spectral side of the Jacquet-Langlands correspondences, and relates the counting function directly to those for the principal congruence subgroups of Eichler orders of level less than one.

This is a joint work with three undergraduate students (Chenhao Tang, Jie Yang and Wenyan Yang) during the event of the 2025 summer school entitled "Algebra and Number Theory" held at the Chinese Academy of Sciences.