山大数论讨论班

星期三，15:00-16:00，2022 春季

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：2022 年春季学期  （星期三，15:00-16:00）

• 2022-03-02，  郗平  （西安交通大学）   21:00-22:00  （注意时间的调整）

  题目：A conjecture of Sárközy on quadratic residues

  摘要：A basic question in additive number theory is to study the sumset A+B for suitably arbitrary sets A,B with some prescribed structures. A conjecture of András Sárközy asserts, for all sufficiently large primes p, that no sumset A+B with |A|,|B| ⩾ 2 consists of all quadratic residues mod p exactly. Sárközy himself proved the ternary analogue of this conjecture, and the original one seems beyond the current techniques. In this talk, we discuss some tight bounds for the possible binary decompositions, which are based on Weil’s bound for complete character sums over finite fields, improving some previous works by I. E. Shparlinski, I. D. Shkredov, and Y.-G. Chen and X.-H. Yan.

  This is joint work with Yong-Gao Chen.

  地点：腾讯会议，ID：358 9878 8585

• 2022-03-09，  齐治  （浙江大学）

  题目：Asymptotic for the Cubic Moment of Maass Form L-Functions

  摘要：In this talk, I will talk about the cubic moment of central L-values for Maass forms. It was studied by Aleksandar Ivić at the beginning of this century, obtaining asymptotic on the long interval [0, T] with error term $O(T^{8/7+\epsilon})$ and Lindelöf-on-average bound on the short window [T-M, T+M] for M as small as $T^{\epsilon}$. Ivić's results are improved in my recent work; in particular, Ivić's conjectured error term $O (T^{1+\epsilon})$ is proven. Our proof follows the standard Kuznetsov--Voronoi approach stemed from the work of Conrey and Iwaniec. Our main new idea is a combination of the methods of Xiaoqing Li and Young.

  地点：腾讯会议，ID：358 9878 8585

• 2022-03-14，  Jesse Thorner  (University of Illinois at Urbana-Champaign)   10:00-11:00

  题目：An unconditional GL(n) large sieve

  摘要：Large sieve inequalities provide powerful notions of "quasi-orthogonality" that are useful in many averaging problems in analytic number theory.  The large sieve for Dirichlet characters is essentially optimal in all aspects, but when trying to extend this result to averages over the full universal family of cuspidal automorphic representations of GL(n) (ordered by analytic conductor), the situation is much more difficult for several reasons.  From 2000-2005, Brumley, Duke,  and Kowalski proved such a large sieve, optimal in the length-aspect, assuming progress towards the generalized Ramanujan conjecture that is only known to hold for the full universal GL(n) family when n is at most 4.  I will present joint work with Asif Zaman in which we remove this hypothesis towards the generalized Ramanujan conjecture, proving an unconditional GL(n) large sieve for all n.  This leads to the first unconditional zero density estimates for the L-functions in this family.  I will also discuss applications to subconvexity bounds.

  地点： Zoom ID: 829 0030 5422,  Password: 202203

• 2022-03-21，  Peter Humphries  (University of Virginia)   10:00-11:00

  题目：Spectral reciprocity and applications

  摘要：Spectral reciprocity is a phenomenon in which certain moments of L-functions are shown to be exactly equal to other moments of L-functions. A quintessential example is Motohashi's formula, which relates the fourth moment of the Riemann zeta function to the third moment of L-functions associated to GL(2) automorphic forms. I will discuss generalisations of Motohashi's formula, how to prove these formulae using tools from the theory of automorphic forms, and applications of these formulae to problems in analytic number theory, including the L^4-norm problem for automorphic forms.

  地点：腾讯会议，ID：358 9878 8585

• 2022-03-30，  方金辉  （南京信息工程大学）  19:00-20:00  （注意时间的调整）

  题目：Representation functions avoiding integers with density zero

  摘要：For a nonempty set A of integers and any integer n, denote r_A (n) by the number of representations of n of the form n=a+a', where a≤a' and a, a'∈A and d_A (n) by the number of pairs (a, a') with a, a'∈A such that n=a-a'. In 2008, Nathanson considered the representation function with infinitely many zeros. As a main result, we prove that, for any set T of integers with density zero, there exists a sequence A of integers such that r_A (n)=1 for all integers n ∉ T and r_A (n)=0 for all integers n∈T, and d_A (n)=1 for all positive integers n. In this talk, we will present our recent results on representation functions.

  地点：腾讯会议，ID：358 9878 8585

• 2022-04-06，  董自康  （Université Paris-Est Créteil, France）

  题目：On large values of the Riemann zeta function
  

  摘要：We will introduce the topic on extreme values of the Riemann zeta function in the half critical strip. On the 1-line, we give a description of the error terms in the result by Aistleitner, Mahatab and Munsch. In the strip 1/2<Re s<1, we improve the result of Bondarenko and Seip in the case σ↘1/2. The improvement is highly depended on the resonancemethod and a new estimate of the GCD sums.

  地点：腾讯会议，ID：358 9878 8585

• 2022-04-13，  千国有  （四川大学）

  题目：On the integers with a divisor in a given interval

  摘要：Let $F(x)$ be an irreducible polynomial with integer coefficients. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. For $F(t)=t$, we write $H_F(x, y,z)$ as $H(x, y,z)$ for simplicity. The estimate for $H(x, y,z)$ is classical and goes back to early work of Besicovitch and Erd\H os in the 1930s. In 2008, Kevin Ford determined the exact order of growth of $H(x,y,z)$ for all $x,y,z$. The corresponding estimate for a linear polynomial $F$ were obtained by Ford and his cooperators using Ford’s method. The study of $H_F(x,y,z)$ for a general polynomial of degree at least 2 began in connection with the problem of bounding from below the largest prime factor of $\prod_{n\le x} F(n)$. In this talk, we will first introduce the story on the investigation of $H(x, y,z)$. And then, we will show heuristic arguments for $H(x, y,z)$ and $H_F(x,y,z)$. Finally, we give some results on the estimate of $H_F(x, y, z)$ obtained by Ford and the speaker.

  地点：腾讯会议，ID：358 9878 8585
  

• 2022-04-20，   Gergely Harcos  (Alfréd Rényi Institute of Mathematics)    16:00-17:00  （注意时间的调整）

  题目：The sequence of prime gaps is graphic

  摘要：Let us call a simple graph on n>1 vertices a prime gap graph if its vertex degrees are 1 and the first n-1 prime gaps (we need the 1 so that the sum of these numbers is even). We can show that such a graph exists for every large n, and under RH for every n>1. Moreover, a sequence of such graphs can be generated by a so-called degree preserving growth process: in any prime gap graph on n vertices, we can find (p_{n+1}-p_n)/2 independent edges, delete them, and connect the ends to a new, (n+1)-th vertex. This creates a prime gap graph on n+1 vertices, and the process never ends. Joint work with P. L. Erdős, S. R. Kharel, P. Maga, T. R. Mezei, and Z. Toroczkai.

  地点：Zoom meeting, ID:  927 3035 6296,   Passcode: 540669

• 2022-04-27，   唐恒才   （河南大学）

  题目：Zero distribution of automorphic L-functions

  摘要：The zero distribution of automorphic L-functions in the critical strip will be discussed. Firstly, we will introduce some results of zeros of Riemann zeta function ζ(s) and Hecke L-function L(s,f). Secondly, the mean value of the argument function of ζ(s) and L(s,f) were considered. By the zero density estimate of L(s,f), the mean value of arg L(s,f) on the critical line in short intervals were given, which improves the work of Sankaranarayanan.

  地点：腾讯会议，ID：358 9878 8585

• 2022-05-04，   刘志新   （天津大学）  14:00-15:00

  题目：华林-哥德巴赫问题的例外集

  摘要：华林-哥德巴赫问题是堆垒素数论的一个重要研究课题。在本次报告中，我们将系统介绍最近二十年华林-哥德巴赫问题例外集的研究进展。

  地点：腾讯会议ID：358 9878 8585

• 2022-05-04，   蒋玉蛟   （山东大学（威海））    15:00-16:00

  题目：Correlations of multiplicative functions

  摘要：In this talk, we shall introduce our work concerning correlations of multiplicative functions, in which one comes from automorphic forms on GL(2). The proof relies on the generalized Bourgain-Katai-Sarnak-Ziegler criterion, Linnik’s dispersion method, sieve method, and analytic theory of automorphic L-functions. As applications, some savings are achieved for shifted convolution problems on  GL(m)× GL(2) (m≥ 4) and Hypothesis C of Iwaniec-Luo-Sarnak for the first time. This is a joint work with Guangshi Lü.

  地点：腾讯会议ID：358 9878 8585

• 2022-05-11，    林永晓   (École Polytechnique Fédérale de Lausanne)

  题目：A result of Xiaoqing Li for self-dual L-functions revisited

  摘要：Let chi_q be real characters of large conductor q. Let f be a fixed GL(2) cusp form of trivial central character. In 2000, by using a moment method, Conrey and Iwaniec obtained the best known subconvexity bounds for self-dual L-functions L(\chi_q,1/2) and L(f\times\chi_q,1/2) in the large q-aspect. This approach was later adapted by Xiaoqing Li to give the first subconvexity bounds for GL(3)xGL(2) self-dual L-functions in the large GL(2) spectral aspect. In this talk, we revisit Li's work and explain how her bounds can be improved to the limit of this method. Some Motohashi-type formulae surrounding these problems will also be discussed. Joint work with Ramon Nunes and Zhi Qi.

  地点：腾讯会议ID：358 9878 8585

• 2022-05-16，   蔡立   （首都师范大学）      10:00-11:00

  题目：Top-degree Ext-groups and Relatively supercuspidal representations

  摘要：Relatively supercuspidal (RSC) representations are the analogue object of supercuspidal representations in the relative Langlands program. In this talk, using top degree Ext-groups via the Schneider-Stuhler duality, we consider the classification of RSC representations in terms of  cuspidal supports and in terms of Arthur parameters for the associated spherical varieties. Especially, we shall focus on the Gan-Gross-Prasad case and the Flicker-Rallis case. Joint work with Yangyu Fan.

  地点：腾讯会议ID：358 9878 8585

• 2022-05-25，  Valentin Blomer  (Universität Bonn)     16:00-17:00

  题目：Uniform Titchmarsh divisor problems

  摘要：The classical Titchmarsh divisor problem asks for the asymptotic evaluation of the divisor function over shifted primes. It is intimately related with primes in long arithmetic progressions. Modern methods can produce strong error terms for fixed shifts, but no progress since the 1960s has been made on the dual problem of summing d(n-p) for p < n or the related problem of Hooley and Linnik of representing a number a sum of a prime and two squares. I will survey approaches and techniques towards the Titchmarsh divisor problem and its variations, and present new results obtained in joint work with Edgar Assing and Junxian Li and with Lasse Grimmelt, Junxian Li and Simon Rydin Myerson. The methods involve a blend of classical analytic number theory, automorphic forms and algebraic geometry.

  地点：Zoom ID: 630 9532 2217,  Passcode: 849782

• 2022-05-30，  Chao Li  (Columbia University)   10:00-11:00

  题目：From sum of two squares to arithmetic Siegel-Weil formulas

  摘要：Can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi and put it in the modern perspective of the Siegel-Weil formula. After illustrating Kudla's influential program on geometric and arithmetic generalizations using the example of modular curves, we discuss recent development on arithmetic Siegel-Weil formulas and applications.

  地点：Zoom ID:  813 9117 8025,   Passcode: 220530

• 2022-06-08，   赵文嘉   （山东大学）

  题目：一类四元二次曲面上殆素数分布问题研究

  摘要：素数分布问题是数论领域的一个重要问题。 哥德巴赫猜想、孪生素数猜想等属于线性的素数分布问题，而一般代数簇上的素数分布理论近些年才有所进展。例如，Brüdern-Dietmann-Liu-Wooley得到了一般性的Birch-Goldbach定理； 再如，对于一般的二次方程，Liu以及Liu-Sarnak研究了其素数解以及殆素数解。在本次报告中, 我们将介绍轨道与素数的一些相关问题与一类四元二次曲面上殆素数分布的问题。

  地点：明德楼 C704