山大数论讨论班

星期三，15:00-16:00，2025年春季学期（3 月开始）

中心校区明德楼 C704

组织者：山大数论组

讨论班着重于数论中的新进展。报告人为校内外学者和学生。欢迎推荐报告。

报告信息（摘要和往年报告信息见列表下方）

• 2025-03-19，Hui Wang 王慧  (山东大学)

  题目：A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms and its application

  摘要：The zero-distribution problem of $L$-functions occupies a central role in analytic number theory as it is intimately connected with various problems involving primes, such as the distribution of primes in short intervals, gaps between primes, and so on. In this talk, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms $\phi$ for $\rm SL(3, \mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions in the spectral aspect. As an application, we obtain an asymptotic formula for the moments of the function $S_\phi(t)=\pi^{-1}\arg L(1/2+it, \phi)$ without assuming the Generalized Riemann Hypothesis (GRH). Previously, such a formula was only known to hold under the GRH. This is a joint work with Professor Qingfeng Sun.

  地点：明德楼 C704

• 2025-03-21，周五， Jing-song Chai   柴劲松   (安徽工程大学)

  题目：Two applications of Bessel functions

  摘要：In this talk, we present two applications of Bessel functions. The first one is to use Bessel functions over finite field to study Asai gamma factors of GL(n). We also obtain a distinction criterion. The second is to use Bessel functions over p-adic field to obtain another Kirillov model for supercuspidal representations.

  地点：明德楼 C704

• 2025-04-02， Didier Lesesvre   (Université de Lille)

  题目：Kuznetsov trace formula for GSp(4) and applications

  摘要：Trace formulas relate statistics on automorphic forms, which often remain mysterious yet central in number theory, with statistics on geometric or arithmetic quantities, which one hopes to be more explicit and better understood. We will discuss how to establish such a Kuznetsov-type trace formula in the case of the symplectic group GSp(4), and will study the precise analytic behaviour of both the spectral and the arithmetic transforms arising in the formula. These fundamental properties can be used to establish various results on the family of Maaß automorphic forms on GSp(4) in the spectral aspect: the Weyl law, a density result on the non-tempered spectrum, large sieve inequalities, bounds on the second moment of the spinor and standard L-functions, as well as a statement on the distribution of the low-lying zeros of these L-functions.

  地点：明德楼 C704

• 2025-04-07，周一， Jun Yu   余君   (北京大学)    10:00-11:00

  题目： Positivity of Fourier transform of zonal spherical functions

  摘要： Given a semisimple real linear group, a zonal spherical function is matrix coefficient associated to the unique spherical vector with value 1 at identity element in a unitary spherical principal series, which are important object in representation theory and harminic analysis. Each zonal spherical function is a positive function. Hence, its Fourier transform along a maximal split torus is everywhere non-negative by a classical theorem of Bochner. In this talk we report a result in a recent joint work: the Fourier transform along a maximal split torus of any zonal spherical function takes positive value everywhere.

  地点：明德楼 C704

• 2025-04-09，  Shigeru Kanemitsu

  题目： Fourier-Whittaker expansion and the Epstein-type zeta-functions

  摘要： As A. Weil described in book ``Dirichlet series and automorphic forms'', E. Hecke made a revolution in the theory of modular forms with his theory of functional equations. His work was continued by his student, H. Maass, which is expounded by R. Bellman. There are some statements about Maass forms being mysterious since they depend on differential equations. We will show that the formulation of Maass' work can be made clear as the Riemann-Hecke-Bochner correspondence between the Maass forms and the ramified functional equations. We elucidate the close relationship with Epstein type zeta-functions, for which Riemann-Hecke-Bochner correspondence takes the simpler form of the Chowla-Selberg integral formula (Fourier-Bessel expansion). We shall give a lot of its important consequences, especially the Lerch-Chowla-Selberg formula, theta transformation formula etc..

  地点： 洪家楼校区1号楼

• 2025-04-16，  Rui Zhang   张蕊   (山东大学)

  题目： 高阶方程在Piatetski-Shapiro素数子集上的Roth型定理

  摘要： Roth定理是加性数论中的核心成果之一，深刻刻画了整数子集在等差数列上的分布性质。近年来，转换原理（transference principle）的引入为这类问题的研究提供了强有力的工具，并推动了一系列重要进展。 本报告将重点讨论如何利用转换原理研究高阶方程在Piatetski-Shapiro素数子集上非平凡解的存在性问题，并介绍我们在该问题上所建立的Roth型定理。

  地点： 明德楼 C704

• 2025-04-23，  Long Jin   金龙   (清华大学)

  题目： Spectral distribution for the twisted Laplacian on hyperbolic surfaces

  摘要： In this talk, we discuss the spectrum of the twisted Laplacian operator on a compact hyperbolic surface. The twisted Laplacian associated with a harmonic form is obtained by conjugating the usual Laplacian-Beltrami operator by an integral of the harmonic form. It is also the Bochner Laplacian associated with the corresponding one-dimensional representation. When the harmonic form is real, the twisted Laplacian is non-self-adjoint but still has discrete spectrum in the complex plane. We will review its spectral theory and connection with the twisted Selberg zeta function. In particular, we show that although most of the spectrum is concentrated near the real axis, there are infinite many eigenvalues away from the real axis, at least when the harmonic form is large enough. This implies the failure of the asymptotic version of Riemann hypotheses for the twisted Selberg zeta function as well as the failure of quantum unique ergodicity. This is joint work with Gong Yulin.

  地点： 明德楼 C704

• 2025-04-28，  Yuk-Kam Lau   刘旭金   (香港大学)   10:00-11:00

  题目： Distribution of the Error Term in the Dirichlet Divsor Problem

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  地点： 洪家楼校区1号楼

• 2025-05-07，  Hengfei Lü   吕恒飞

  题目： The relative Langlands Program

  摘要： Given a local field F, define a reductive G over F. Suppose that there exists an involution on G such that the fixed points form another reductive group, denoted by H. Then G/H is a symmetric pair. How to describe the space S(G/H), i.e., those smooth representations Pi of G restricted to H containing the trivial character as a quotient, is so called the relative Langlands program. They are related to the nonvanishing property of certain L-functions at some special points. In this talk, we will give a brief introduction, including the Langlands functoriality conjecture, the multiplicities problems and the special value of certain L-functions, based on the examples. In particular, we will focus on the Galois involution and the PTB pairs.

  地点： 明德楼 C704

• 2025-05-14，  Yakun Xi   席亚昆

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  地点： 明德楼 C704

• 2025-05-28，  Lingmin Liao   廖灵敏

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  地点： 明德楼 C704

• 2025-06-04，   

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• 2025-06-11，  

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• 2025-06-18，  

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往年报告：

• 2024年秋季报告    2024年春季报告   
  

• 2023年秋季报告    2023年春季报告

• 2022年秋季报告 2022年春季报告

• 2021年秋季报告 2021年春季报告

• 2020年秋季报告 2020年春季报告

Contact us at: brhuang "at" sdu dot edu dot cn,   Office: Mingde C701,   Tel: 883-69786