山大数论讨论班

星期二，14:00-15:00，2020年

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年秋季学期    （星期二，14:00-15:00）

• 2020年9月8日，Daniel El-Baz（Graz University of Technology）

  题目：A pair correlation problem and counting lattice points via the zeta function

  摘要：The pair correlation function is a local measure of the randomness of a sequence. The behaviour of the pair correlation of sequences of the form ({a_n alpha}) for almost every real number alpha where (a_n) is a sequence of integers is by now relatively well-understood. In particular, a connection to additive combinatorics was made by relating that behaviour to the additive energy of the sequence (a_n).

    Zeev Rudnick and Niclas Technau have recently started investigating the case of (a_n) being a sequence of real numbers. This talk is based on joint work in progress with Christoph Aistleitner and Marc Munsch in which we pursue this line of research.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年9月15日，吕广世（山东大学）

  题目：On Shifted Convolution Sums of Arithmetic Functions

  摘要：In this talk, we shall summarize our recent results on various shifted convolution sums of arithmetic functions. We develop some simple approaches to study correlations of Fourier coefficients of cusp forms with other arithmetic functions. Our results improve and streamline some previous results established by more sophisticated methods, and establish some new results on shifted convolution sums for higher rank groups.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年9月22日，赵立璐（山东大学）

  题目：小素数乘积在算术级数中的分布

  摘要：设 (a,q)=1 且 q 充分大. Erdos-Oldyzko-Sarkozy 猜想断言存在素数 $p_1,p_2<q$ 使得 $p_1p_2\equiv a\pmod{q}$. 本报告简介在该课题上的一些研究结果. 例如，Walker 证明了当 q 为充分大的素数时，存在素数 $p_1,\ldots,p_{48}<q$ 使得 $p_1\cdots p_{48}\equiv a\pmod{q}$. 报告中将介绍筛法应用中的奇偶性障碍.

  地点：明德楼 C702

• 2020年9月29日，孟宪昌（University of Göttingen）  延期进行

  题目：Distinct distances on hyperbolic surfaces

  摘要：Erdos in 1946 asked the question of finding the minimal number of distinct distances among any $N$ points in the plane. In our work, we give complete answer for this problem on all hyperbolic surfaces of finite volume. For any cofinite Fuchsian group $\Gamma\subset\mathrm{PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $\Gamma\backslash\mathbb{H}^2$ determines $\geq C_{\Gamma} \frac{N}{\log N}$ distinct distances for some constant $C_{\Gamma}>0$ depending only on $\Gamma$. In particular, for $\Gamma$ being any finite index subgroup of $\mathrm{PSL}(2, \mathbb{Z})$ with  $\mu=[\mathrm{PSL}(2, \mathbb{Z}): \Gamma ]<\infty$,  any set of $N$ points on $\Gamma\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{\mu\log N}$ distinct distances for some absolute constant $C>0$.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年10月6日，王世纶 （帕多瓦大学）

  题目：An introduction to Rankin--Selberg L-function and its applications

  摘要：Rankin and Selberg introduced a new tool into the study of cusp forms independently at around the same time, which is known today as the Rankin--Selberg method. In 1985, Waldspurger got an important formula of the Rankin--Selberg L function. After that, Yuan, Zhang, Zhang generalized the Waldspurger’s formula, and get a similar formula on Shimura curves. Later, Cai, Shu, Tian established the most general explicit version of the Waldspurger formula. In this talk, I will give an overall introduction to the Rankin--Selberg L function. Also I start with an elliptic curve which is associated with the cube sum problem and give some arithmetic applications of the explicit Waldspurger formula. 

  地点：明德楼 C702

• 2020年10月15日，王标 （SUNY Buffalo） 10:00-11:00 （注意报告时间的不同）

  题目：Analogues of Alladi's formula

  摘要：In this talk, we will mainly introduce the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the M\"{o}bius function $\mu(n)$, which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. Several examples will be given. For instance, if $(k, \ell)=1$, then

      $$-\sum_{\begin{smallmatrix}n\geq 2\\ p(n)\equiv \ell (\operatorname{mod} k)\end{smallmatrix}}\frac{\mu(n)}{\varphi(n)}=\frac1{\varphi(k)},$$

  where $p(n)$ is the smallest prime divisor of $n$, and $\varphi(n)$ is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, We will give some conjectures on more analogues.

  地点：腾讯会议，会议 ID：995 338 269
  

• 2020年10月22日，易少云 （University of South Carolina）  10:00-11:00 （注意报告时间的不同）

  题目：An equidistribution theorem for cuspidal automorphic representations for GSp(4)

  摘要：Equidistribution theorems for a family of automorphic representations of a reductive group have been studied in various aspects by many mathematicians. This is connected to the equidistribution of Hecke eigenvalues of classical modular forms. In this talk we will discuss an equidistribution result, a version of so-called automorphic Plancherel density theorem, for a family of cuspidal automorphic representations of GSp(4). We formulate our theorem explicitly in terms of the number of cuspidal automorphic representations of GSp(4) satisfying certain conditions at the local places. To count the number of these cuspidal automorphic representations, we will explore the connection between Siegel modular forms of degree 2 and cuspidal automorphic representations of GSp(4). This is a joint work with Manami Roy and Ralf Schmidt.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年10月27日，杨李扬 (California Institute of Technology)

  题目：Average Central L-values on U(2,1)$\times$ U(1,1), Nonvanishing and Subconvexity

  摘要：In this talk, we study an average of automorphic periods on $U(2,1)\times U(1,1)$. We also compute local factors in Ichino-Ikeda formulas for these periods to obtain an explicit asymptotic expression. Combining them together we would deduce some important properties of central L-values on $U(2,1)\times U(1,1)$ over certain family: the first moment, nonvanishing and subconvexity. This is joint work with Philippe Michel and Dinakar Ramakrishnan.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年11月03日，Ofir Gorodetsky （Oxford）

  题目：The distribution of squarefree integers in short intervals

  摘要：The squarefree integers are divisible by no square of a prime. It is well known that they have a positive density within the integers. We consider the number of squarefree integers in a random interval of size H: # {n in [x,x+H] : n squarefree}, where x is a random number between 1 and X. The variance of this quantity has been studied by R. R. Hall in 1982, obtaining asymptotics in the range H < X^{2/9}, with a proof method that stays in 'physical space'. Keating and Rudnick recently conjectured that his result persists for the entire range H < X^{1-epsilon}. We make progress on this conjecture, with properties of Dirichlet polynomials playing a role in our results. We will show how one can verify the conjecture for H slightly beyond X^{1/2}. This is joint work with Kaisa Matomäki, Maks Radziwill and Brad Rodgers.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年11月10日，张德瑜（山东师范大学）

  题目：On the index of composition of integral ideal

  摘要：In this talk, we firstly introduce some properties of Dedekind zeta function, then as an application we discuss  some results about  the  index of composition of ideal.  Finally we show some  recent progress about this question. This is joint work with Wenguang Zhai.

  地点：明德楼 C702

• 2020年11月17日，翟文广（中国矿业大学（北京））

  题目：On a generalization of the Euler totient function

  摘要：J. Kaczorowski defined the generalized Euler totient function $\varphi(n, F)$ corresponding to a  polynomial Euler product $F$. Let $E(x, F)$ denote the error term in the asymptotic formula of the summatory function of $\varphi(n, F).$  J. Kaczorowski proved that the mean square of $E(x, F)$ has an asymptotic formula if $F$ satisfies GRH. We can prove a sharper asymptotic formula under GRH.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年11月24日，路志鹏（University of Göttingen）

  题目：Erdos distinct distances problem in hyperbolic surfaces

  摘要：Erdos distinct distances problem asks for the lower bound of number of distinct distances between pairs of points from any finite point sets of given size. The problem in the Euclidean plane was revolved by L. Guth and N. H. Katz in 2011. We study the problem in hyperbolic surfaces. The key in our work is to introduce an invariant (we call it "geodesic cover") for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of this invariant to study the distinct distances problem in hyperbolic surfaces. Especially, for S from a large class of hyperbolic surfaces, we establish the nearly optimal bound >=C_S N/\log N for distinct distances determined by any N points in S, where C_S>0 is some constant depending only on S. In particular, for S being modular surface or standard regular of genus >=2, we evaluate C_S explicitly.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年12月01日，孟宪昌（University of Göttingen）

  题目：Distinct distances on hyperbolic surfaces

  摘要：Erdős in 1946 asked the question of finding the minimal number of distinct distances among any $N$ points in the plane. In our work, we give complete answer for this problem on all hyperbolic surfaces of finite volume. For any cofinite Fuchsian group $\Gamma\subset\mathrm{PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $\Gamma\backslash\mathbb{H}^2$ determines $\geq C_{\Gamma} \frac{N}{\log N}$ distinct distances for some constant $C_{\Gamma}>0$ depending only on $\Gamma$. In particular, for $\Gamma$ being any finite index subgroup of $\mathrm{PSL}(2, \mathbb{Z})$ with  $\mu=[\mathrm{PSL}(2, \mathbb{Z}): \Gamma ]<\infty$,  any set of $N$ points on $\Gamma\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{\mu\log N}$ distinct distances for some absolute constant $C>0$.

  地点：腾讯会议，会议 ID：995 338 269

• 2020年12月08日，Ezra Waxman (Technische Universität Dresden)  15:00-16:00 （注意报告时间的不同）

  题目：Random Models for Artin Twin Primes

  摘要：We say that a prime number p is an Artin prime for g if g is a primitive root mod p.  For appropriately chosen integers g and d, we present a conjecture for the asymptotic number of prime pairs (p,p+d) such that both p and p+d are Artin primes for g.  Our result suggests that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.  (Joint work with Magdaléna Tinková and Mikuláš Zindulka).

  地点：Zoom 会议，Zoom ID：527 845 3958

• 2020年12月15日，王梦迪（山东大学）

  题目：Recent developments on polynomial Szemeredi configuration

  摘要：In 1996, Bergelson and Leibman proved that any positive density subset of [N] contains a non-trivial progression of the form x, x+P_1(y),…,x+P_k(y), where P_1,…,P_k\in\mathbb{Z}[y] are polynomials with zero constant terms. This is the first polynomial generalization of Szemeredi’s theorem. In this talk, I am going to discuss some developments on polynomial arithmetic progressions, and this is based on recent works of Sarah Peluse and Sean Prendiville, and upcoming works of Sean Prendiville, Xuancheng Shao and myself.

  地点：明德楼 C702