山大数论讨论班

星期三，16:30-17:30，2021 秋季

中心校区明德楼 C702，济南

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：2021年秋季学期

• 2021-09-15， 王志伟  （山东大学）

  题目：The twin prime conjecture and twin friable conjecture

  摘要：In this talk，we will talk about the developments on the twin prime conjecture and twin friable conjecture. This talk is based on joint work with Yujiao Jiang and Guangshi Lü.

• 2021-09-22， 无报告。 中秋节。

• 2021-09-29， 无报告。 山大齐鲁青年论坛。

• 2021-10-06， 王子豪  （山东大学）

  题目：A Bombieri-Vinogradov theorem of higher rank groups

  摘要：The celebrated Bombieri-Vinogradov theorem states that the primes up to x in arithmetic progressions modulo q are well-distributed for all q≤x^{1/2} (logx)^{-B}, which shows that the GRH is true on average. In this talk, we present an unconditional generalization of Bombieri-Vinogradov theorem in the GL(n) automorphic context over number fields. In particular, we give the same quality as the result of Bombieri-Vinogradov when n≤4. Moreover, we improve the level of distribution comparing to previous conditional results. As applications, we also discuss some shifted convolution problems at integers and primes. If time permits, I will introduce some ingredients of the proof. This is a recent joint work with Yujiao Jiang, Guangshi Lü, and Jesse Thorner.

• 2021-10-13， 无报告。 校庆。   （报告调整到 10 月 21 日）

• 2021-10-20， 吴小胜 （合肥工业大学）

  题目：On zeros of Dirichlet L-functions

  摘要：We will introduce the classical Levinson's method, and the process of the study of zeros of Dirichlet L-functions on the critical line. Some processes on simple zeros are also enclosed.

  腾讯会议，会议 ID： 358 9878 8585

• 2021-10-21， Dorian Goldfeld  (Columbia University)     Time: 10:00-11:00 am

  Title: The template method for Langlands Eisenstein series on GL(n)

  Abstract: A template is a pattern used as a guide to produce similar things. This talk will explain how Borel Eisenstein series (whose Fourier coefficients are well known and have been studied for over 50 years) can be used to determine Fourier coefficients of Langlands Eisenstein series. This is joint work with Steven D. Miller and Michael Woodbury.

  Zoom Meeting ID: 959 4778 8710,   Passcode: 384896

• 2021-10-27， Yuk-Kam Lau  (The University of Hong Kong)

  Title: Distribution of the trace of Frobenius of elliptic curves over a finite field

  Abstract: In this talk we are concerned with the trace of Frobenius of elliptic curves over a finite field, which is the fluctuation of the number of points on the elliptic curve over the finite field deviated from the expected value. We shall survey some recent results and their connection with other objects in the proofs. In particular, we shall look into the work of Bringmann, Kane and Pujahari in the arXiv.

  腾讯会议，会议 ID： 358 9878 8585

• 2021-11-03， Stephen Lester  (King's College London)

  Title: Lattice points on hyperbolic circles

  Abstract: Given a point  in the complex upper half plane and discrete subgroup  of   , the hyperbolic lattice point problem is to determine the number of -translates of  that lie within a hyperbolic circle as the radius grows. This may be viewed as a non-Euclidean analogue of the Gauss circle problem. In this talk I will give an overview of some results on the hyperbolic lattice point problem and will also present some recent work concerning the angular distribution of lattice points lying on hyperbolic circles. This is joint with Dimitrios Chatzakos, Pär Kurlberg, and Igor Wigman.

  Zoom 会议，会议 ID：853 8434 2184,   Passcode: 211103

• 2021-11-08、11-15、11-22、11-29，  郗平 （西安交通大学）  周一上午 9:30-11:00

  短期课程：筛法不器

  摘要：随着对孪生素数猜想、哥德巴赫猜想等著名问题的研究，筛法逐渐发展为一个古典又富有生命力的数论方法。我们试图简明地介绍筛法的基本框架、基础理论及其在素数分布问题中的具体应用，主要包括历史背景、Selberg筛法、线性组合筛法、Henryk Iwaniec关于二次多项式表殆素数的工作、张益唐关于相邻素数间隔的工作等，重点阐述筛法的显赫与无奈。若时间允许，或许也介绍一点筛法的几个“非传统”应用，如Kloosterman和的等分布、代数簇上的有理点分布等。

  腾讯会议，会议 ID： 358 9878 8585

• 2021-11-10， 孙庆峰  （山东大学）

  题目：Analytic twists of GL(2)×GL(2) automorphic forms

  摘要：In this talk we will talk about some results on analytic twists of GL(2)×GL(2) automorphic forms with applications. This is a joint work with Bingrong Huang and Huimin Zhang.

  
  

  腾讯会议，会议 ID： 358 9878 8585

• 2021-11-17， Sa'ar Zehavi  (Tel Aviv University)

  Title: Sectorial equidistribution of the roots of x^2=-1 mod p

  Abstract: The equation x2 + 1 = 0 mod p has solutions whenever p = 2 or 4n+1. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a famous theorem of Duke, Friedlander and Iwaniec from 1995. We examine what happens to the distribution when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. This simple arithmetic question has a solution which involves multiple disciplines of number theory, but the talk does not assume any previous background.

  Zoom meeting ID： 840 3136 8437,   Passcode: 211117

• 2021-11-24，何晓光  （四川大学）

  题目：Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L-functions

  摘要：In this talk, we introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic L-function on GL_m (m \geq 2) and polynomial nilsequences has logarithmic decay; the other is that the mean value of the M\"obius function, coefficients of automorphic L-function and polynomial nilsequences also has logarithmic decay.

  腾讯会议，会议 ID： 358 9878 8585

• 2021-12-01， 无报告。

• 2021-12-05， 山大数论日。  报告人：

      李红泽（上海交通大学）

      孙智伟（南京大学）

      王天泽（华北水利水电大学）

      吴杰（CNRS & UPEC）

  周日 9:20-11:40， 14:30-16:40

• 2021-12-08， 无报告。 PANT conference

• 2021-12-15， 陈永高  （南京师范大学）

  题目：埃及分数

  摘要：分子为 1 的分数称为单位分数或埃及分数. 将有理数表成特定分母的埃及分数之和是一个古老的研究课题，本报告将介绍相关的进展. 特别地，将介绍把 1 表成埃及分数之和的有关结果，其中这些埃及分数的分母具有给定的素因子.

  腾讯会议，会议 ID： 358 9878 8585

• 2021-12-22， 高帆  （浙江大学）     时间：17:00-18:00

  题目：A problem on the size of theta representations

  摘要：Study of the automorphic representations of linear algebraic groups constitutes the core of the Langlands program. For central covers of such groups, their genuine representations illustrate several challenging aspects of the theory. In this talk, we will give a brief (and inevitably biased) exposition on the background and context of genuine representation theory, and eventually concentrate on a problem regarding the “simplest” family of representations of covers - the theta representations.

  腾讯会议，会议 ID： 358 9878 8585