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