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