山大数论讨论班

星期三，10:00-11:00，2024年春季学期（4 月下旬开始）

中心校区明德楼 C702

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）

注：2024 年 1 月至 4 月 在瑞典 Institut Mittag-Leffler 有 Analytic Number Theory Program. 几乎每周都有四个解析数论方向的报告，可以线上参加。

• 2024-04-24，Yuk-Kam Lau (刘旭金)   (香港大学)    09:00-10:00

  题目：Pseudorandomness in Kloosterman Sums

  摘要：A pseudorandom sequence is a sequence that looks random but is actually generated by a deterministic process. In this talk we will discuss the pseudorandomness underlying the Kloosterman sums motivated by Mok ancobserved by Fouvry, Michel, Rivat and Sarkozy respectively. In particular, we will look at their measurements of pseudorandomness and their results.

  地点：明德楼 C702

• 2024-04-24，Chan Heng Huat (曾衡发)   (National University of Singapore)
  

  题目：Modular equations and series for 1/π

  摘要：In this talk, I will explain how to derive series for 1/π published by Ramanujan in his article ``Modular equations and approximations to π''. Modular forms, modular equations, hypergeometric series, as well as Hilbert class fields of imaginary quadratic fields all play an important role in these derivations.

  地点：明德楼 C702

• 2024-05-07，Igor Wigman  (King's College London)   10:00-11:00

  题目：The Robin problem on rectangles

  摘要：We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum, depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, we show that the pair correlation function of the Robin spectrum on a Diophantine rectangle is Poissonian. This talk is based on a joint work with Zeev Rudnick.

  地点：明德楼 C702

• 2024-05-08，Igor Wigman  (King's College London)

  题目：Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus

  摘要：This talk is based on a joint work with Zeev Rudnick.

  We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. We show that in this same limit, the energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature.

  地点：明德楼 C702

• 2024-05-17，Zhizhong Huang (黄治中)   (AMSS)   Friday  09:00-10:00

  题目：The Manin--Peyre conjecture for a family of Fano threefolds with quadratic fibrations

  摘要：We establish an asymptotic formula for the number of rational points of bounded anticanonical height outside of a thin subset on the bi-projective variety $V$ defined by $$L_1(x_1,x_2)y_1^2+L_2(x_1,x_2)y_2^2+L_3(x_1,x_2)y_3^2+L_4(x_1,x_2)y_4^2=0$$ in $\mathbb{P}^1\times \mathbb{P}^3$, where $L_i,1\leqslant i\leqslant 4$ are pairwise independent linear forms. This settles the thin set version of the Manin--Peyre conjecture for $V$. The proof uses a mixture of the circle method and techniques from the geometry of numbers. This is joint work with Dante Bonolis and Tim Browning.

  地点：明德楼 C702

• 2024-05-17，Lasse Grimmelt   (University of Oxford)   Friday  10:00-11:00

  题目：Twisted Correlations of the divisor function via discrete averages of Poincare series

  摘要：The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these problems are addressed by reducing them to sums of Kloosterman sums, followed by either the use of existing black-box results or by-hand application of spectral theory through Kuznetsov's formula.

  In this presentation, based on joint work with Jori Merikoski, I will introduce an alternative approach that entirely circumvents the need for Kloosterman sums. This approach offers increased flexibility compared to existing black-box methods. It is inspired by Bruggeman-Motohashi's work on the fourth moment of the zeta function. As an application, I will present a novel result on correlations of the divisor function in arithmetic progressions and moments of Dirichlet L-functions.

  地点：明德楼 C702

• 2024-05-22，Yuchao Wang (王玉超)   (上海大学)

  题目：On almost prime points on unirational varieties

  摘要：Sarnak and his collaborators initiated a program to investigate the distribution of points whose coordinates have few prime factors on varieties equipped with a group structure. In this talk, we shall concentrate on the case of unirational varieties. We prove that there exists an integer $r$ such that rational points for which the product of the coordinates has at most $r$ prime factors form a Zariski dense subset, provided that the unirational variety has one rational point. Moreover, we may obtain rather small bounds for $r$ for several special cases. The proof relies on the geometric structure of the varieties and the use of analytic tools.

  地点：明德楼 C702

• 2024-05-29，Liangxun Li (李良汛)   (山东大学)

  题目：Multiplicative functions in short intervals and the second moment of $L$-functions

  摘要：In this talk， we will discuss two methods in proving Matom\"aki--Radziwi\l\l's theorem. The first method is based on the Vinogradov--Korobov zero-free region for $\zeta$-function. And the second method is based on the Hal\'asz's theorem. Motivated by these proof ideas, under the generalized Ramanujan conjecture, we obtain a small log-saving on the second integral moment of $L(1/2+it, \pi)$ where $\pi$ is an irreducible cuspidal automorphic representation of $GL_d(\mathbb{A})$ ($d\geq 3$). Specifically， the bound $$\int_{T}^{2T}\Big|L\big(\frac{1}{2}+it, \pi\big)\Big |^2 d t\ll_{\pi} \frac{T^{\frac{d}{2}}}{\log^{\eta}T}$$ holds for any positive constant $\eta\leq \frac{1}{400d^4}$. In addition, assuming the new zero-free region for $L$-functions, we can also obtain a small log-saving on the second moment of $L(1/2, \pi\times \chi)$ in $q$-aspect. This is a joint work in progress with Haozhe Gou.

  地点：明德楼 C702

• 2024-06-05，No meeting

  (Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact)

• 2024-06-12，Nawapan (Ploy) Wattanawanichkul   (University of Illinois Urbana-Champaign)

  题目：Holomorphic quantum unique ergodicity and weak subconvexity for L-functions

  摘要：Quantum unique ergodicity (QUE) describes the equidistribution of the L2-mass of eigenfunctions of the Laplacian as their eigenvalues approach infinity. My focus lies on a specific variant known as holomorphic QUE,  which concerns the distribution of the L2-mass of normalized Hecke eigenforms of even weight k (where k ≥ 2). In 2010, Soundararajan and Holowinsky proved the equidistribution of normalized Hecke eigenforms as k tends to infinity. In my talk, I will discuss their proof ideas, explore the connection with the subconvexity problem, and present my new results on the topic. 

  地点：Voov meeting ID: 358 9878 8585

• 2024-06-19，Wenbin Zhu  (齐鲁工业大学)

  题目：Some problems about friable numbers in Piatetski-Shapiro sequences

  摘要：In this talk, we will show several results about friable numbers in Piatetski-Shapiro sequences, such as the ternary Goldbach type result, Diophantine approximation, almost primes, intersections of Piatetski-Shapiro sequences and so on. These results were established mainly by two exponential sums involving fractional powers over friable numbers. The proofs of these exponential sums rely on the method of double sums, the exponential sums over integers and some properties of friable numbers.

  地点：明德楼 C702

• 2024-06-27，Bin Chen  (根特大学)    Thursday  10:00-11:00

  题目：Small gaps between almost-twin primes

  地点：明德楼 C702

往年报告：

• 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

山大数论讨论班

星期三，15:00-16:00，2023年秋季学期

中心校区明德楼 C702 或者 线上

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）

• 2023-09-13，Winston Heap （Norwegian University of Science and Technology） 

  题目：Mean values of long Dirichlet polynomials

  摘要：We survey some applications of mean value results for Dirichlet polynomials over primes in the theory of the Riemann zeta function. This includes  central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for moments of long Dirichlet polynomials over primes without using the Hardy--Littlewood conjectures for additive correlations of the von-Mangoldt functions.

  地点：明德楼 C702

• 2023-09-20，董自康 （同济大学）

  题目：Distribution of Large Character Sums

  摘要：We study the distribution of large quadratic character sums based on the recent work of Lamzouri, we obtain the structure results of quadratic characters with large character sums which generalize the previous work of Bober, Goldmakher, Granville and Koukoulopoulos to the family of quadratic characters.

  地点：明德楼 C702

• 2023-10-11，Shigeru Kanemitsu (金光滋)   (Kyushu Institute of Technology)

  题目：Chebyshev polynomials, fundamental units; Dirichlet convolutions

  摘要：In this talk, we present two topics. One is an elucidation of the Richaud-Degert fundamental units in the real quadratic field $\mathbb{Q}(\sqrt{D})$, where $D=c^2+r$, $c$ is a positive integer and $r=\pm 1, \pm 4$. We reveal the underlying structure in terms of Chebyshev polynomials (standard and of hyperbolic type) in relation to Pisot numbers.

      The other is about an asymptotic formula for the convolution of the PNT-related function and the Piltz divisor function. This is a good example of the hyperbola method and reveals the contribution of the error terms $\sqrt{x}\delta(x)$ of PNT and $\sqrt{x}^{\alpha_\kappa+\varepsilon}$, $\alpha_\kappa\le \frac{\kappa-1}{2\kappa}￥ of the Piltz divisor problem. These are cowork with R. –Y. Li.

  地点：明德楼 C702

• 2023-10-18，魏斌（天津大学）

  题目：On large values of the Riemann zeta function

  摘要：The Riemann zeta function is a fundamental topic in number theory. In this talk, our primary focus will be on the behavior of extreme values with the zeta function. We will first provide an introduction to the research history, covering various aspects as the behavior on the 1-line, within the critical strip, and on the critical line. We will also include results of the derivatives of the zeta function. Then we will present some recent results. The talk is mainly based on some joint work with Zikang Dong.

  地点：明德楼 C702

• 2023-10-25，Takao Komatsu (小松尚夫)   (浙江理工大学)

  题目：Diophantine problems of the equations x^2+y^2=z^r  

  摘要：As the Pythagorean triple (s^2-t^2,2 s t,s^2+t^2), the equation x^2+y^2=z^r has the parametrizations for r\ge 2. By using them, we could give the convergents of certain values by rationals x/y such that x, y satisfy Diophantine equations. We can also give the Frobenius numbers formed from Diophantine equations. In this talk we consider the equations x^2+y^2=z^r for r=2,3,4,5.  

  地点：明德楼 C702

• 2023-10-28，李金蒋（中国矿业大学(北京)）   周六   10:00-11:00

  题目：On Chen’s theorem over Piatetski-Shapiro type primes and almost-primes

  摘要：In this talk, we establish a new mean value theorem of Bombieri-Vinogradov's type over Piatetski-Shapiro sequence. As an application of the mean value theorem, for $\gamma$ satisfying

     \[  1-\frac{0.03208}{2^{38}+17}<\gamma<1,  \]

  we prove that there exist infinitely many primes p such that p+2=P_2 with P_2 being Piatetski-Shapiro almost-primes of type $\gamma$, and there exist infinitely many Piatetski-Shapiro primes p of type  $\gamma$ such that p+2=P_2. These results generalize the result of Pan and Ding, and constitute an improvement upon a series of previous results of Li-Zhang-Xue, Wang-Cai, Peneva, etc.

  地点：明德楼 C702
  

• 2023-11-01，Steve Lester (King's College London)    10:00-11:00

  题目：Around the Gauss circle problem

  摘要：Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-R disc by its area is O(R^{1/2+o(1)}). One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square-root cancellation of sums of these random variables. In this talk I will examine this heuristic and discuss how lattice points near the circle interact with one another. This is joint work with Igor Wigman.

  地点：明德楼 C702

• 2023-11-01，Steve Lester (King's College London)

  题目：Spacing statistics for lattice points on circles

  摘要：In this talk I will describe the distribution of lattice points lying on circles. A striking result of Kátai and Környei shows that along a density one subsequence of admissible radii the angles of lattice points lying on circles are uniformly distributed in the limit as the radius tends to infinity. Their result goes further, proving that uniform distribution persists even at very small scales, meaning that the angles are uniformly distributed within quickly shrinking arcs. A more refined problem is to understand how the lattice points are spaced together at the local scale, e.g. given a circle containing N lattice points determine the number of gaps between consecutive angles of size less than 1/N.  I will discuss joint work with Pär Kurlberg in which we compute the nearest neighbor spacing of the angles along a density one subsequence of admissible radii.

  地点：明德楼 C702

• 2023-11-08，王世纶   (Università degli studi di Padova)

  题目：Explicit reciprocity law for finite slope families of modular forms

  摘要：Recent years, Andreatta and Iovita construct a new triple product p-adic L-function for the finite slope families of modular forms. Combining with the diagonal class constructed by Bertolini, Seveso and Venerucci, we can establish the relation between the new p-adic L-function and the diagonal class in the finite slope case. I will give a brief talk on the proof of the reciprocity law and also talk about some applications. This is a work in progress. 

  地点：明德楼 C702

• 2023-11-15，温婷婷   (山东大学)

  题目：The number of rational points on a class of singular cubic hypersurfaces

  摘要：Let Q(y) be any ternary positive definite quadratic form with integral coefficients. The equation x^3 = Q(y)z represents a class of singular cubic hypersurfaces. In this talk, we mainly introduce the quantitative behaviour of rational points on these hypersurfaces, and describe the ideas, methods, and some related results. This is motivated by Manin's conjecture, which predicts the asymptotic formula of rational points on algebraic varieties. This is a joint work with Yujiao Jiang and Wenjia Zhao.

  地点：明德楼 C702

• 2023-11-22，No Talk

• 2023-11-29，No Talk

• 2023-12-06，No Talk

• 2023-12-13，户亚青  (山东大学)

  题目：Polynomial Maps and Polynomial Sequences in Groups

  摘要：We review some functional definitions of polynomial maps on abelian groups, present a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and prove that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. This is based on a work recently accepted by Journal of Group Theory.

  地点：明德楼 C702

• 2023-12-20，李文威  (北京大学)

  题目：Towards a Shimura-Waldspurger correspondence for nontempered spectra

  摘要：Using theta lifting, Gan-Ichino proved a multiplicity formula for the tempered automorphic spectrum of the adelic metaplectic group Mp(2n, A), which generalizes Waldspurger's celebrated work when n=1, at least in the tempered case. Based on the recently obtained stable trace formula for Mp(2n, A), I will sketch how can this be possibly extended to the whole automorphic spectrum, leading up to a multiplicity formula of Arthur type. This is a project in early stage, and I will only explain the basic ideas.

  地点：明德楼 C702

• 2023-12-25， 刘博辰    (南方科技大学)   10:00-11:00

  题目：Some examples of number theory in analysis

  摘要：This talk surveys number theory that appeared in my research on analysis. We shall first introduce the Erdos distance problem in which the integer lattice serves as the sharp example. Then we will discuss its continuous version, the Falconer distance problem, in which the sharp example is given by Jarnik's construction based on diophantine approximation. In the second half of this talk we shall introduce tiling and multiple tiling by translations in the Euclidean space, and explain how counting solutions of linear equations in the multiplicative group influences our result on the multiplicity of tiling.

  地点：明德楼 C702

• 2023-12-27，张晓宇  (Universität Duisburg-Essen)  10:00-11:00

  题目：Mod p theta liftings for unitary groups

  摘要：Theta liftings are explicit transfers of automorphic forms between two classical groups and play an important role in Langlands program. In this talk, we consider an arithmetic (mod p) version of theta liftings from a compact unitary group to a split unitary group. We construct an explicit theta series which transfers an automorphic form to a non-zero mod p automorphic form under mild conditions. In the end of the talk, we would like to give some applications of the above result.

  地点：明德楼 C702

• 2024-01-24，Cédric Pilatte   (University of Oxford)    16:00-17:00

  题目：Recent progress on binary correlations of the Liouville function

  摘要：In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set S of positive integers that is both a Sidon set and an asymptotic basis of order 3. This means that the sums of two elements of S are all distinct, while the sums of three elements of S cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.

  地点：腾讯会议  Voov meeting ID: 822-3376-8649,    Password: 123456

往年报告：

• 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

山大数论讨论班

星期二，15:00-16:00，2023年

中心校区明德楼 C702 或者 线上

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）

暑期专场

• 2023-02-28，曹阳 （中国科学技术大学）   10:00-11:00

  题目：Relative version of weak approximation for abelian schemes

  摘要：Over number field, weak approximation for algebraic varieties is about the density of rational points in adelic points. In this talk, for an abelian scheme over a rational variety, instead of considering the weak approximation, we consider the relative closed subgroup generated by rational points in adelic points of the total space, and study its relation with cohomological obstruction.

  地点：明德楼 C702

• 2023-02-28，刘华锋 （山东师范大学）

  题目：The values of general quadratic forms at prime arguments

  摘要：Let f_1, …, f_r be regular indefinite integral quadratic forms with s prime variables. In this talk, we discuss the values taken by real linear combinations of f_1, …, f_r. It is proved that, under certain conditions, the values of the real linear combinations of three general quadratic forms, with 6 prime variables in total, can approximate any real numbers. The previous results of this type focus on diagonal forms.

  地点：明德楼 C702

• 2023-03-07，李晓承   (北京大学)

  题目：An estimate for spherical functions on SL(3,R)

  摘要：We prove an estimate for spherical functions $\varphi_\lambda(a)$ on SL(3,R), establishing uniform decay in the spectral parameter $\lambda$ when the group variable $a$ is restricted to a compact subset of the abelian subgroup $A$. In the case of SL(3,R), it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when $\lambda$ and $a$ vary.

  地点：明德楼 C702

• 2023-03-14，Number Theory Pi Day   12:00-17:00

  地点：明德楼 C702

• 2023-03-21，Matthew P. Young   (Texas A&M University)   21:00-22:00

  题目：Large sieve inequalities for families of L-functions

  摘要：Large sieve inequalities are useful and flexible tools for understanding families of L-functions.  The quality of the bound is one measure of our understanding of the corresponding family.  For instance, they may directly give rise to good bounds for certain moments of L-functions.  In this talk, I will survey this problem and discuss recent progress for some new families.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-03-30，Xiannan Li   (Kansas State University)   10:00-11:00

  题目：Quadratic twists of modular L-functions

  摘要：The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young.  Our proof depends on deriving an optimal large sieve type bound.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-04，NO TALK (Qingming Festival)

• 2023-04-11，Max Wenqiang Xu  徐文强   (Stanford University)  14:00-15:00

  题目：Recent progress in random multiplicative functions

  摘要：Random multiplicative functions are probabilistic models for arithmetic functions that number theorists care about. In this talk, I  will survey the subject and describe some recent progress as well as some open problems. 

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-18，Winston Heap

  题目：Simultaneous extreme values of zeta and L-functions

  摘要：We use a modification of the resonance method to demonstrate simultaneous large values of L-functions on the critical line. The method extends to other families and can be used to show both simultaneous large and small values. Joint work with Junxian Li. 

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-21，郭振宇   (西安交通大学)   15:00-16:00

  题目： The distribution of consecutive special primes under the Hardy-Littlewood conjecture

  摘要： In this talk, we study the distribution of pairs of consecutive Beatty or Piatetski-Shapiro primes and give conjectures with the prime counting functions. We give a heuristic argument to support the predictions which relies on a strong form of the Hardy-Littlewood conjecture. The method of the argument is highly related to Chebyshev’s bias and exponential sums.

  地点：明德楼 C702

• 2023-04-25，陈昌昊   (安徽大学)

  题目：Measure theory of Weyl sums

  摘要：I will first introduce the Weyl sums and its applications on number theory, PDE and ergodic theory. Then I will talk about the typical asymptotic bounds of Weyl sums in the sense of measures and Baire category. Joint works with R. Baker, K. Bryce, J. Maynard and I. Shparlinski.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-05-02，Matthew de Courcy-Ireland   (Stockholm University)

  题目：Six-dimensional sphere packing and linear programming

  摘要：This talk is based on joint work with Maria Dostert and Maryna Viazovska. We prove that the Cohn--Elkies linear programming bound is not sharp for sphere packing in dimension 6. This is in contrast to Viazovska's sharp bound in dimension 8, even though it is believed that closely related lattices achieve the optimal densities in both dimensions. The proof uses modular forms to construct feasible points in a dual program, generalizing a construction of Cohn and Triantafillou to the case of odd weight and non-trivial Dirichlet character. Non-sharpness of linear programming is demonstrated by comparing this dual bound to a stronger upper bound obtained from semidefinite programming by Cohn, de Laat, and Salmon. Our construction has vanishing Fourier coefficients along an arithmetic progression, which can be explained using skew self-adjointness of Hecke operators.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-05-09，李时璋   (中国科学院)

  题目：On the cohomology of classifying stacks

  摘要：我们将从有限平坦交换群概型说起，这些对象由迪厄多内模所分类，我们将看到如何从这些群概型的分类叠的上同调中得到迪厄多内模，最终我们将陈述一些关于这些分类叠的上同调的较为一般的结构性定理：这是我在与Kubrak以及Mondal的合作中得到的一个结果。

  地点：明德楼 C702

• 2023-05-16，华晟昊   (山东大学)

  题目： The distribution of central values of quadratic twisted L-functions

  摘要： We introduce some recent progress on distribution of central values of $L$-functions in quadratic twists families, including the determination of GL(3) automorphic forms, the Keating--Snaith conjecture, and the Farmer--Gonek--Hughes conjecture. This talk is based on joint works with Bingrong Huang and a joint work in progress with Mingyue Fan and Sizhe Xie.

  地点：明德楼 C702

• 2023-05-23，杨鹏宇   (中国科学院)

  题目：The distribution of periodic torus orbits in homogeneous spaces

  摘要：The distribution of periodic torus orbits in homogeneous spaces is related to several problems in number theory, such as distribution of integral points on spheres and distribution of CM points on modular curve. In this talk, we first recall the history of this subject, and then give an entropy bound for limit distributions of periodic torus orbits in the case of $\rm PGL_4$. Joint work with A. Wieser.

  地点：明德楼 C702

• 2023-05-30，Claire Burrin   (University of Zurich)

  题目：Pairs of saddle connections

  摘要：There is a collection of classical problems at the intersection of number theory and analysis that concern the distribution of lattice or primitive lattice points in their ambient space. In two dimensions, we can think of the set of primitive integer vectors in the plane as describing the set of primitive simple closed geodesics (up to free homotopy) on the two-dimensional torus. In this talk I will explain how counting saddle connections on Veech surfaces stands as a natural ‘higher genus’ analogue to the classical primitive lattice point counting problem. I will then discuss a recent result (in joint work with Jon Chaika and Samantha Fairchild) on counting pairs of saddle connections.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-06-06，黄治中   (中国科学院)

  题目： 二次超曲面有理点的分布

  摘要： 作为经典Hasse—Minkowski定理的一个实效版本，我们将利用Heath-Brown发展的delta圆法给出二次超曲面上满足适当局部条件的有理点增长率的估计，并讨论最优误差项与丢番图逼近之间的联系。本次报告基于与 D. Schindler (Göttingen) 和 A. Shute (Bristol) 最近的合作工作。

  地点：明德楼 C702

• 2023-06-13，王好武   (武汉大学)    15:00-16:00

  题目：The classification and construction of singular-weight automorphic products

  摘要：Automorphic products are modular forms with infinite product expansions and special zeros. For example, the denominator of the Weyl-Kac character formula of any affine Lie algebra defines an automorphic product of singular (i.e. minimal) weight as a Jacobi form. In this talk, we discuss automorphic products of singular weight on orthogonal groups $\rm O(n,2)$. This type of modular form is often the denominator of some Borcherds-Kac-Moody Lie algebra, and is related to vertex operator algebras. It was conjectured that there are only finitely many such modular forms. We present some results on the classification and construction of these exceptional modular forms. This talk is based on joint work with Brandon Williams.

  地点：明德楼 C702

• 2023-06-13，吴涵   (中国科学技术大学)    16:00-17:00

  题目：Some uniform Weyl subconvex bounds based on distributional Motohashi's formula

  摘要：We presented a distributional Motohashi's formula at this number theory seminar 3 years ago. We will briefly present a refined/published version of this formula. Based on this version, applications to the uniform Weyl-type subconvex bounds for some $PGL(2)$ $L$-functions, as well as the limitation of the method, will be discussed. These results are joint works with Balkanova--Frolenkov and Ping Xi, respectively. If time permits, we will present a further application to the partition function, joint work with Nicholas Andersen.

  地点：明德楼 C702

  
  

• 2023-06-16，肖玄玄   (澳门科技大学)    09:00-10:00

  题目：Large Values of L-functions and the Resonance Method

  摘要：In this report, we give an introduction of the resonance method. The method can be used in the study of large values of L-functions. With the design of the resonator, we give new results for Dirichlet L-functions and Rankin-Selberg L-functions.

  地点：明德楼 C702

• 2023-06-16，魏志宁   (俄亥俄州立大学)    10:00-11:00

  题目：The refined strong multiplicity one and its applications

  摘要：In this talk, I will introduce several refined strong multiplicity one theorems for Siegel paramodular forms for symplectic groups. As an applicationI will give a criterion for determining a Siegel cusp form by twisted L-values. This is a joint work with Xiyuan Wang, Pan Yan and Shaoyun Yi.

  地点：明德楼 C702

• 2023-07-03，张瑞祥   (美国加州大学伯克利分校)    09:00-10:00

  题目：A stationary set method for estimating oscillatory integrals

  摘要：Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

  地点：明德楼 C702

• 2023-07-03，杨道道   (格拉茨工业大学)    10:00-11:00

  题目：Variants of GCD sums and extreme values of derivatives of the Riemann zeta function

  摘要：GCD sums naturally appear in a Diophantine approximation problem considered by Hardy and Littlewood in 1922.  It was until 1949 that  Gál solved their  problem.  Since  Gál's fundamental theorem, there have been new developments in the past decade. I will review these results and report some new results on the limit of GCD sums and the log-type GCD sums.   Another topic of the talk concerns large values of derivatives of zeta and L-functions. Large values of  zeta and L-functions are classical topics in analytic number theory, which can be dated back to a result of Bohr and Landau in 1910. Resonance methods are modern tools to produce large values of zeta and L-functions. GCD sums are one of important ingredients. I will talk on producing large values of derivatives of zeta and L-functions via resonance methods. The link between  log-type GCD sums and derivatives of the Riemann zeta function will also be discussed.

  地点：明德楼 C702

• 2023-08-08，侯家齐  (University of Wisconsin-Madison)  08:30-09:20

  题目：Periods of eigenfunctions on arithmetic hyperbolic 3-manifolds

  摘要：Let $X$ be a compact congruence arithmetic hyperbolic 3-manifold, and $f$ a Hecke--Maass form on $X$ with spectral parameter tending to infinity. In this talk, I will present a bound for the period of $f$ along a hyperbolic surface in $X$, which improves the local bound. This result is based on the method of arithmetic amplification.

  地点：明德楼 C702

• 2023-08-08，孙宇宸  (图尔库大学)  09:40-10:30

  题目：On divisor bounded multiplicative functions in short intervals

  摘要：In 2016, the celebrated Matomaki--Radziwi\l\l theorem showed that  there are cancellations for 1-bounded multiplicative functions in almost all short intervals. Our recent work proved that Matomaki--Radziwi\l\l theorem can be extended to divisor bounded multiplicative functions. Especially, we proved that for any sufficiently large $X$, $\epsilon>0$ and $h \geq (\log X)^{(1+\epsilon)k \log k- k +1}$, we have

       $$\frac{1}{h} \sum_{x<n \leq x+h}d_k(n) - \frac{1}{x}  \sum_{x<n \leq 2x}d_k(n) = o(\log^{k-1}x)$$

  for almost all $x \in [X,2X]$, where $d_k(n) = \sum_{m_1 \cdots m_k = n} 1$.

  地点：明德楼 C702

• 2023-08-08，Max Wenqiang Xu 徐文强  (Stanford University)  10:40-11:30

  题目：Positive definite quadratic characters

  摘要：We call a quadratic character $\chi$ mod $d$ positive definite if its partial sums at all different truncations are nonnegative. It is known that such characters are rare due to an earlier result of Baker, Montegomery and a recent result of Kalmynin.  We give a new upper bound, which is conjectured to be close to optimal based on a probabilistic model. The proof is based on establishing a $L$-function analogue of Fydorov--Hiary--Keating conjecture.

  This is joint work in progress with R. Angelo and K. Soundararajan.

  地点：明德楼 C702

往年报告：

• 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

山大数论讨论班

星期三，15:00-16:00，2022年

线上会议  或者  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）

Evening Section: 

• 2022-09-07，  王六权 （武汉大学）

  题目：Sign changes of coefficients of powers of the infinite Borwein product

  摘要：We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\leq 24$. We  provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.

  地点：腾讯会议ID：358 9878 8585

• 2022-09-14，  沈权利 （University of Lethbridge）

  题目：The eighth moment of the Riemann zeta function

  摘要：The first and second moments of the Riemann zeta function were established asymptotically by Hardy-Littlewood and Ingham, respectively. It is believed that sixth and higher moments are beyond current techniques. Ng recently proved an asymptotic formula for the sixth moment by assuming a conjecture on additive divisor sums. In this talk, we will discuss about how to compute the eighth moment under a conjecture on additive divisor sums and the Riemann hypothesis. It is a joint work with Nathan Ng and Peng-Jie Wong.

  地点：腾讯会议ID：358 9878 8585

• 2022-09-23 (周五), 10:00-11:00,  刘青阳 （中国科学院）

  题目：On Random Multiplicative Functions

  摘要：Random multiplicative functions were first introduced by Wintner as a probabilistic model for the Mobius function and have attracted quite a lot of attention in number theory and probability. In this report, we will introduce the development and recent research of the random multiplicative function theory, and briefly describe the probabilistic methods behind it. Finally, we will present the reporter's contribution in this direction. 

  地点：腾讯会议ID：358 9878 8585

• 2022-09-28，  王梦迪 （KTH Stockholm）

  题目：Linear equations in smooth numbers

  摘要：In this talk, I will introduce an approach to counting system of linear equations with smooth number variables, and this is based on joint work with Lilian Matthiesen. I plan to avoid introducing formal notation in higher-order Fourier analysis, hope this will be accessible to a fairly general audience.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-05， 无报告 （国庆节）

• 2022-10-12， 栗慧曦 （南开大学）

  题目：Two results related to Pan’s work

  摘要：In 1979 Pan proved a new mean value theorem and applied it to prove Chen’s theorem. In the first part of the talk we will apply this theorem to the representation of a large integer as ap + bP_2. In 1982 Pan made a new attempt on the Goldbach conjecture. In the second part of the talk we will discuss how his approach connects the Goldbach conjecture to equidistribution of some arithmetic functions.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-19，  无报告

• 2022-10-26，  吴云辉 （清华大学）    14:00-15:00

  题目：Recent progress on first eigenvalues of hyperbolic surfaces for large genus

  摘要：In this talk we wil discuss several recent results on first eigenvalues of closed hyperbolic surfaces for large genus. For example, we show that a random hyperbolic surface of large genus has first eigenvalue greater than $\frac{3}{16}-\epsilon$, extending Mirzakhani's lower bound $0.0024$. This talk is based on several joint works with Yuhao Xue.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-26，  李加宁 （山东大学）

  题目：Genus理论的一个技巧和应用

  摘要：给数域的循环扩张, Genus理论描述了大域的理想类群Galois不变部分的大小. 它由Gauss发现并用此研究二次域的2理想类群. 本报告将介绍一个构造辅助循环扩张并嵌套使用genus理论的技巧, 并用此证明当素数p =4,7 mod 9时, Q(sqrt[3]{p})的类数恰好被3整除, 这证明了Barrucand-Cohn, Lemmermeyer等人关于纯三次域类群和单位群的猜想. 这是与张神星的合作工作. 

  地点：腾讯会议ID：358 9878 8585

• 2022-11-02，  安金鹏 （北京大学）

  题目：丢番图逼近中的奇异性与齐性动力系统

  摘要：对于一个实数，如果Dirichlet逼近定理中的常数可以任意小，则称该实数具有奇异性。容易证明，这样的实数即为有理数，所以这一定义并无实质意义。但是，对向量或矩阵引入类似的定义后，人们发现奇异向量或奇异矩阵的集合具有丰富的分形性质。报告人将探讨具有奇异性的数、向量和矩阵的性质及其与齐性动力系统的关系，并介绍在联立奇异性方面与人合作的一项工作。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-09，  肖梁  （北京大学）

  题目：模形式的p进斜率及相关问题

  摘要：模形式是朗兰兹纲领中的重要研究对象之一，一方面它与椭圆曲线和伽罗华表示的算术性质紧密联系，另一方面它固有的解析性质为研究椭圆曲线提供了巨大的帮助。这个报告将首先介绍模形式相关的背景和结果之后慢慢引入模形式的p进性质的研究，特别是对特征模形式系数被p整除次数（即p进斜率）的性质的研究。这一类研究始于90年代Gouvea和Mazur大量的数值计算，他们对特征模形式的p进斜率提出若干猜想。之后Coleman-Mazur, Buzzard-Calegari, Bergdall-Pollack逐步从理论上完善关于p进斜率的这些猜想。如果时间允许，我会汇报在这个方面与刘若川、万大庆、赵斌、Nha Truong的系列合作，在某些情况下证明关于p进斜率的这些猜想。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-16，  张涵  （清华大学）

  题目：Nondivergence of reductive group actions on homogeneous spaces

  摘要：Let $G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $G/\Gamma$. The question we are interested in is whether there is a compact set of $G/\Gamma$ that intersects every H-orbit.

  We show that the failure of this can be explained by a single algebraic reason, which generalizes several previous results towards this question. We also obtain a way to find this algebraic obstruction, if there is any. 

  This talk is based on joint work with Runlin Zhang.

  地点：腾讯会议ID：358 9878 8585

• 2022-11-23，   刘若川 （北京大学）

  题目：p进簇的黎曼-希尔伯特问题

  摘要：我们将介绍p进簇的黎曼-希尔伯特问题及一些相关的进展。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-23，  Ritabrata Munshi   (Indian Statistical Institute)   19:00-20:00
  

  Title:  GL(3) L-functions on the critical line

  Abstract:  Traditionally moments of L-functions on the critical line have been used to bound the size of the L-functions on the critical line. For example the Weyl bound for the Riemann Zeta function can be concluded either by obtaining a strong bound for the error term in the second moment or by bounding short fourth moment. The same method works in the case of degree two L-functions, although computing the moment becomes a tricky business. Till recently we had no similar result for degree three (or higher) L-functions. This talk will be about t-aspect sub convexity for degree three L-functions - the delta method approach, its limitations and the recent blending of the moment method and the delta method (a joint work with Aggarwal and Leung). 

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-11-30，   马蔷   （山东大学）  15:00-16:00

  题目：Exponential sums with the Dirichlet coefficients of Rankin-Selberg L-functions

  摘要：In this talk, we shall introduce our recent work concerning cancellation  inadditively twisted sums  on $\mathrm{GL}(m)\times\mathrm{GL}(m^{\prime})$. We describe a new method to obtain upper bounds for exponential sums with multiplicative coefficients without the Ramanujan conjecture. We verify these hypothesis for (with mild restrictions) the Rankin-Selberg $L$-functions attached to two cuspidal automorphic representations. 

  地点：腾讯会议ID：358 9878 8585

• 2022-11-30，  Péter Maga   (Alfréd Rényi Institute of Mathematics)   20:00-21:00
  

  Title: On the sup-norm problem of automorphic forms: new counting methods

  Abstract: As it was proved by Sarnak, the supnorm of eigenfunctions of the Laplacian on a compact symmetric Riemannian manifold can be estimated from above by an appropriate power (given in terms of some invariants of the space) of their Laplace eigenvalue. Examples show that Sarnak's exponent is sharp in some cases. However, when the space has also arithmetic symmetries (i.e. Hecke operators) and we restrict to joint eigenfunctions of the Laplacian and the Hecke operators, one might expect a better exponent. Such power-savings are known for arithmetic quotients of several symmetric spaces. In the talk, I will give a brief overview of the history of the problem, and discuss some new matrix counting methods which might potentially yield the desired power-saving in some currently unsettled cases. The talk will be based on ongoing research (joint with Gergely Zábrádi).

  Venue: Zoom: 825 9309 8083, password: 2022
  

• 2022-12-07，   曹炜   （闽南师范大学）

  题目：Divisibility on point counting over finite Witt rings

  摘要：Let F_q denote the finite field of q elements with characteristic p. Let Z_q denote the unramified extension of the p-adic integers Z_p with residue field F_q. In a joint work with Prof. Daqing Wan, we study the q-divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Z_q/p^mZ_q, where the n variables of the polynomials are restricted to run through a combinatorial box lifting F_q^n. We prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmuller box. This extends the classical Ax-Katz theorem over finite field F_q (the case m=1). Taking q=p to be a prime, our result extends and improves a recent combinatorial theorem of Grynkiewicz. Our different approach is based on the addition operation of Witt vectors and is conceptually much more transparent.

  地点：腾讯会议ID：358 9878 8585

• 2022-12-14，   Sheng-Chi Liu  (Washington State University)   13:00-14:00

  题目：A GL(3) Analog of Selberg's Result on S(t)

  摘要：The function S(t) appears in an asymptotic formula for counting the number of nontrivial zeros of the Riemann zeta function with imaginary part less than t. It was shown by Littlewood that the function has a lot of cancellation on the average over t. Later Selberg studied the moments of S(t) and the moments of the analog function associated with a Dirichlet L-function for a primitive Dirichlet character. A GL(2) analog of Selberg's result was proved by Hejhal and Luo. In this talk we will discuss a GL(3) analog of such results. This is joint work with Shenhui Liu.

  地点：Zoom: 825 9309 8083, password: 2022

• 2022-12-14，  Min Lee   (Bristol University)   20:00-21:00
  

  Title:  Non-vanishing of symmetric cube L-functions

  Abstract:  The non-vanishing of L-series at the centre of the critical strip has long been a subject of great interest. An important example of the significance of non-vanishing is in the case of an L-series corresponding to a modular form of weight 2, where the non-vanishing at the central point has been shown to be equivalent to the finiteness of the group of rational points of the associated elliptic curve. In the case of higher rank L-functions whose Euler product has an even degree, such connections between non-vanishing at the central point and the finiteness of certain groups are believed to be true, but the relations remain purely conjectural. The symmetric cube L-series plays a role in one of these conjectures.

  Ginzburg, Jiang and Rallis (2001) proved that the non-vanishing at the central point of the critical strip of the symmetric cube L-series of any GL(2) automorphic form is equivalent to the non-vanishing of a certain triple product integral. The main purpose of this talk is to use this equivalence to prove that there are infinitely many Maass-Hecke cuspforms over the imaginary quadratic field of discriminant -3 such that the central values of their symmetric cube L-functions do not vanish.

  This is joint work with Jeff Hoffstein and Junehyuk Jung. 

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-12-21，   郑维喆  （中科院）

  题目：平展上同调和超积

  摘要：平展上同调的超积提供了代数簇的一族Weil上同调理论，其性质与ℓ进上同调的ℓ无关性和无挠性密切相关。本报告将介绍超积上同调研究的新进展。

  地点：腾讯会议ID：358 9878 8585

• 2022-12-21， Lior Bary-Soroker   (Tel Aviv University)     20:00-21:00
  

  Title:  Rational points coming from ramified covers

  Abstract:  The talk aims to present recent progress in the area Hilbert's irreducibility theorem.

  The theorem may be formulated as the statement there are "many" rational points on the *line* not coming from rational points on a given cover. If one replaces the line by other varieties, the situation becomes more complicated and, in particular, there are obstructions to the theorem coming from rational points of unramified covers (recall that the line is simply connected).

  I will present recent two results: The first result, j/w Daniele Garzoni, deals with affine groups: We show that a random walk on a finitely generated Zariski dense subgroup almost surely misses rational points coming from a ramified cover. The second result, j/w Arno Fehm and Sebastian Petersen, deals with abelian varieties and with rational points over the maximal cyclotomic field (or more generally, the field obtained by adding the torsion points of an abelian variety).

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-12-26，  Rizwanur Khan  (University of Mississippi)

  Title: The fourth moment of Dirichlet L-functions and related problems

  Abstract: I will discuss asymptotics for the fourth moment of Dirichlet L-functions and related problems, especially with regards to simplifying existing approaches and sharpening the error terms in these asymptotics. 

  Venue: Zoom: 825 9309 8083, password: 2022

往年报告：

• 2022年春季报告  

• 2021年秋季报告   2021年春季报告

• 2020年报告

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

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

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

山大数论讨论班

星期三，15:00-16:00，2021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：2021年春季学期  （星期三，15:00-16:00）

• 2021年03月02日，Jacques Benatar (Tel Aviv University) （注意报告时间的不同）

  题目：Moments of random trigonometric polynomials with multiplicative coefficients

  摘要：In this talk I will discuss some recent work, joint with Alon Nishry and Brad Rodgers, concerning the distribution of trigonometric polynomials

  Q_f,N(θ) = ∑_n≤N f(n) e(nθ).

  We will first discuss some results and conjectures for deterministic sequences f(n) and then move on to random coefficients f(n) = X(n) which are multiplicatively generated by a sequence of Rademacher or  teinhaus random variables. I will give a Salem-Zygmund type result for the distribution of Q_X,N and explain how it relates to an arithmetic point counting problem, via a moment method. Obtaining almost sure estimates for ||Q_X,N||_∞ presents additional challenges.

  腾讯会议，会议 ID：995 338 269

• 2021年03月10日，管彬 （山东大学）

  题目：Relative trace formula and averages of central values of triple product L-functions

  摘要：Feigon and Whitehouse studied central values of triple product L-functions averaged over newforms of weight 2 and prime level. They proved some exact formulas applying the results of Gross and Kudla which link central values of triple L-functions to classical "periods". In this talk, I will generalize their results for more cases using Jacquet's relative trace formula and Ichino's period formula, and apply these average formulas to the nonvanishing problem. 

  地点：明德楼C702

• 2021年03月17日，翟帅 （University of Cambridge）

  题目：Classical Diophantine problems and the Birch and Swinnerton-Dyer conjecture

  摘要：The Birch and Swinnerton-Dyer conjecture is one of the principal open problems of number theory today, which asserts that the size of the group of rational points on an elliptic curve is related to the behavior of the central value of the associated L-function. In this lecture, I will present a few classical Diophantine problems, and their connection to the conjecture of Birch and Swinnerton-Dyer.

• 2021年03月24日，魏菲 （清华大学）

  题目：Disjointness of Moebius from rigid dynamical systems

  摘要：In this talk, I will introduce some results about Sarnak’s Moebius Disjointness Conjecture for rigid dynamical systems, and the estimate involved in the average value of the Moebius function in short arithmetic progressions.

  腾讯会议，会议 ID：995 338 269

• 2021年03月31日，于树澄 （Uppsala University）

  题目：A quantitative Khintchine-Groshev Theorem with congruence conditions

  摘要：Given a non-increasing approximating function $\psi$, the classical Khintchine-Groshev Theorem gives a zero-one law on the Lebesgue measure of the set of $\psi$-approximable matrices. In this talk we will discuss a quantitative Khintchine-Groshev Theorem with an extra congruence condition. Our method replies on a classical lattice point counting result of Schmidt. This is joint work with Mahbub Alam and Anish Ghosh.

  腾讯会议，会议 ID：995 338 269

• 2021年04月07日，高鹏 （北京航空航天大学）

  题目：Bounds for moments of cubic and quartic Dirichlet L-functions

  摘要：Moments of L-functions at the central point over a family of characters of a fixed order have been extensively studied in the literature. Although many results are available on moments of quadratic Dirichlet L-functions, less is known for the moments of higher order Dirichlet L-functions. In this talk, I will explain how to apply the recent lower bounds principal of Heap and Soundararajan and an earlier method of Soundararajan with a refinement by Harper to obtain sharp bounds for moments of central values of cubic and quartic Dirichlet L-functions.  This is a joint work with L. Zhao.

  腾讯会议，会议 ID：995 338 269

• 2021年04月14、21、28日，无报告。（线下有数论方向短期课程。）

• 2021年05月05日，无报告。五一假期。

• 2021年05月12日，吴峙佑 （Max Planck Institute for Mathematics） 15:00 - 17:00

  题目：P-adic Hodge, prisms and Galois representations

  摘要：P-adic Hodge theory is an analogue of classical Hodge theory in complex geometry, it plays an important role in the study of Galois representations, which is fundamental in algebraic number theory. Recently, Peter Scholze and Bhargav Bhatt found a new perspective on p-adic Hodge theory, namely the theory of prisms. In a recent preprint, we prove that Galois representations and (φ,Γ)-modules can both be incorporated in this new framework, and we can recover the classical result that these two objects are equivalent. In this talk, I will explain this result together with the general background.

• 2021年05月19日，李俊贤 （Max Planck Institute for Mathematics）

  题目：Uniform Titchmarsh divisor problems

  摘要：The Titchmarsh divisor problem asks for an asymptotic evaluation of the average of the divisor function evaluated at shifted primes. We will discuss how strong error terms that are uniform in the shift parameters could be obtained using spectral theory of automorphic forms. We will also discuss the automorphic analogue of the Titchmarsh divisor problem. This is a joint work with E. Assing and V. Blomer.

  腾讯会议，会议 ID：995 338 269

• 2021年05月26日，无报告。硕士博士毕业论文答辩。

• 2021年06月02日，胡广伟 （山东大学）

  题目：The circle method and shifted convolution sums

  摘要：In this talk，we will consider some shifted convolution sums involving the Fourier coefficients of theta series. Here some approaches are devoted to generalize and improve previous results, and our results do not depend on the Ramanujan conjecture. This is a joint work with Guangshi Lü.

• 2021年06月09日，尹洪波 （山东大学）

  题目：Borcherds Product and CM Value of Lambda function

  摘要：In this talk, we will talk about Gross-Zagier's type factorization formulas for the CM value of special modular functions using the Borcherds product and Bruinier-Kudla-Yang's big CM value formula.  We will illustrate the method using the Lambda modular function which is a joint work with Tonghai Yang and Peng Yu. 

• 2021年06月16日和17号，齐治 （浙江大学）  时间：16日 16:00-17:00,  17日 10:00-11:00

  题目：Some results on the central L-values for GL(2) and GL(3)

  摘要：In these talks, I will talk about the moments of central L-values for GL(2) Maass forms and the subconex problem for self-dual GL(3) Maass forms over imaginary quadratic fields. Our main tools are the Kuznetsov trace formula for GL(2) and Voronoi summation formulae for GL(2) and GL(3) over number fields. My focus will be on the analysis on the complex numbers. These are joint works with Sheng-chi Liu.

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

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