山大数论讨论班

星期一，15:00-16:00，2020年春季

中心校区明德楼C702，济南

组织者：刘建亚 和 黄炳荣

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：2020年春季学期 （周一 15:00-16:00）

• 2020年4月27日，Zeev Rudnick (Tel Aviv University) 15:00-16:00

  题目：Prime lattice points in ovals

  摘要：The study of the number of lattice points in dilated regions has a long history, with several outstanding open problems. In this lecture, I will describe a new variant of the problem, in which we study the distribution of lattice points with prime coordinates. We count lattice points in which both coordinates are prime, suitably weighted, which lie in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. We obtain an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term. Time permitting, I will explain some background motivation coming from Quantum Chaos. (joint work with Bingrong Huang)

  网络会议，加入Zoom会议，会议ID：635 813 7757

• 2020年5月04日，吴杰（CNRS & UPEC） 15:00-17:00 （注意报告时间的不同）

  题目：让我们与素数共舞

  摘要：数论是数学的一个重要分支并有着丰富的理论与广泛的应用。数论的核心是研究素数的分布及解丢番图方程，本报告将焦注前者。Landau 在1912年剑桥的ICM大会上的报告里，从众多关于素数的猜想中选出四个：

          1. 哥德巴赫猜想

          2. 孪生素数猜想

          3. 勒让德猜想

          4. 欧拉猜想

  我们将围绕着这四大猜想，通过详细地介绍相关的历史与现状，来理解其中深刻的思想内含和丰富的方法技巧。让我们与素数共舞，领会这四大猜想如何引领百年来的素数研究，进一步提高大家对数论研究的兴趣。

  网络会议，加入Zoom会议，会议ID：635 813 7757（这个号不可用） 790 719 5930 （暂时使用）

• 2020年5月11日，王英男（深圳大学） 15:00-15:50 （注意报告时间和会议ID的不同）

  题目：On the exceptional set of the generalized Ramanujan conjecture  

  摘要：For any prime $p$ and Hecke-Maass form $\phi$ on $GL(n) (n\geq2)$, $\alpha_{\phi,1}(p),\ldots,\alpha_{\phi,n}(p)$ denote the corresponding Satake parameters of $\phi$ at $p$. The generalized Ramanujan conjecture asserts that $|\alpha_{\phi,i}(p)|=1$, $i=1,\ldots,n$. In this talk we will survey the recent results and developments centered on this problem. This is a joint work with Lau Yuk-Kam and Ng Ming Ho.

  网络会议，加入Zoom会议，会议ID：678 547 5442 

• 2020年5月11日，肖玄玄（澳门科技大学） 16:10-17:00 （注意报告时间和会议ID的不同）

  题目：Circle Method in Roth Problem and in Thin Subgroup of $SL_2(\mathbb{C})$

  摘要：Circle method is an important tool in classical Waring-Goldbach problems. The first part of the report aims to introduce an application in classical Roth Problems.

  In the past decade, Bourgain and Kontorovich studied integers from thin group orbits using circle method. The second part of the report aims to introduce an asymptotic local-global principle problem in thin subgroup of $SL_2(\mathbb{C})$. 

  Circle method in these two problems will be compared at the end.

  网络会议，加入Zoom会议，会议ID：678 547 5442 
  

• 2020年5月18日，郗平（西安交通大学） 15:00-16:00 （注意我们使用腾讯会议）

  题目：A weighted Selberg sieve and applications
  

  摘要：Selberg invented a weighed sieve by divisor functions towards the twin prime conjecture in 1950’s. We will outline this old and powerful idea, as well as the subsequent applications to the prime-tuple conjecture. Moreover, we will also discuss recent applications to the distribution of Kloosterman sums.

  腾讯会议，会议 ID：995 338 269

• 2020年5月18日，吴涵（Queen Mary University of London） 16:15-17:15 （注意报告时间和会议ID的不同）

  题目：On Motohashi’s formula
  

  摘要：The original Motohashi's formula relates the fourth moment of the Riemann zeta function to the cubic moment of the L-functions attached to modular forms of full level. Based on various integral representations of L-function, Michel-Venkatesh gave a beautiful "one-line proof" of Motohashi's formula, recently rigorously implemented by Nelson. Based on Weil's re-interpretation of integral representations, we will offer a "one-graph" proof in this talk. We may discuss the relation with other methods if time permits.

  腾讯会议，会议 ID：995 338 269
  

• 2020年5月25日，齐治（浙江大学） 15:00-16:00

  题目：Bessel functions and Beyond Endoscopy

  摘要：In this talk, I will first introduce the thesis of Akshay Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ $L$-functions on $\mathrm{GL}_2$ over $\mathbb{Q}$ or a totally real field. The idea follows a suggestion of Peter Sarnak on using the Kuznetsov relative trace formula instead of the Arthur-Selberg trace formula for the Beyond Endoscopy problem. I will then discuss how to generalize Venkatesh's work from totally real to arbitrary number fields. The main supplement is an integral formula for the Fourier transform of Bessel functions over $\mathbb{C}$.

  腾讯会议，会议 ID：995 338 269

• 2020年6月01日，梁兵兵（IMPAN） 10:00-11:00 （注意报告时间的不同）

  题目：Mean dimension and von Neumann-Lück rank

  摘要：The mean dimension is an invariant of topological dynamical systems, which is crucial to solve the embedding problems of dynamical systems. The von Neumann-Lück rank is an invariant of group ring modules in $L^2$-invariants theory. We show that the mean dimension of every algebraic action, coincides with the von Neumann-Lück rank of the associated modules. The crucial ingredient of the proof is the introduction of a new invariant for integral group ring modules.

  腾讯会议，会议 ID：995 338 269

• 2020年6月08日，叶扬波（The University of Iowa） 9:00-10:30 （注意报告时间和会议ID的不同）

  题目：Techniques in Number Theory

  网络会议，加入Zoom会议，会议ID：995 2657 8586 

• 2020年6月08日，王丹（山东大学） 15:00-16:00

  题目：The Selberg–Delange method in short intervals for the Dedekind zeta function

  摘要：Many number theoretic problems lead to the study of the mean values of arithmetic functions. Between 1954 and 1971, Selberg and Delange developed a quite general method using the analytic properties of the Dirichlet series associated to the arithmetic function. This is nowadays known as the Selberg-Delange method. In this talk, we establish a general mean value result of Dedekind Zeta-function over short intervals with the Selberg-Delange method.

  腾讯会议，会议 ID：995 338 269

• 2020年6月15日，张伟（山东大学） 15:00-16:00

  题目：On cube-free numbers of the form [n^c] over special sequences and their applications

  摘要：In this talk, we will prove that there are infinite cube-free numbers in some special Piatetski-Shapiro sequences of the form [n^c] with fixed c(1<c<2). This improves previous results. This can also be seen as a generalization for the result of Deshouillers. As an application, we also get lower bounds for certain sums with fixed c(1<c<2). Previously, such type lower bounds were only given with fixed c(1<c<149/87) by Baker, Banks, Brudern, Shparlinski and Weingartner. Such type lower bounds can be used to deal with the largest prime factor for Piatetski-Shapiro sequences.

  腾讯会议，会议 ID：995 338 269