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Bingrong Huang

## Personal Information

Professor   Supervisor of Doctorate Candidates   Supervisor of Master's Candidates

# NTSeminar

### Orgonizers: SDU Number Theory Group (SDU-NTG)

The seminar will discuss topics in current research in number theory. Presentations will be by students, locals and visitors. Welcome to suggest talks.

See HERE.

See HERE.

### Schedule: Spring semester 2021

• Mar. 02, 2021. Jacques Benatar (Tel Aviv University)   (Note the special time.)

Title: Moments of random trigonometric polynomials with multiplicative coefficients

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

Qf,N(θ) = nN 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 Steinhaus random variables. I will give a Salem-Zygmund type result for the distribution of QX,N and explain how it relates to an arithmetic point counting problem, via a moment method. Obtaining almost sure estimates for ||QX,N|| presents additional challenges.

Tencent VooV Meeting, ID：995 338 269

• Mar. 10, 2021. Bin Guan (Shandong University)

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

Abstract: 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.

Venue: Mingde Building C702

• Mar. 17, 2021. Shuai Zhai (University of Cambridge)

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

Abstract: 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.

• Mar. 24, 2021. Fei Wei (Tsinghua University)

Title: Disjointness of Moebius from rigid dynamical systems

Abstract: 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.

Tencent VooV Meeting, ID：995 338 269

• Mar. 31, 2021. Shucheng Yu (Uppsala University

Title: A quantitative Khintchine-Groshev Theorem with congruence conditions

Abstract: 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.

Tencent VooV Meeting, ID：995 338 269

• Apr. 07, 2021. Peng Gao (Beihang University)

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

Abstract: 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.

Tencent VooV Meeting, ID：995 338 269

• Apr. 14/21/28, 2021. No talks. (There are some minicourses on number theory during this time.)

• May 05, 2021. No talk. May Day Holiday.

• May 12, 2021. Zhiyou Wu (Max Planck Institute for Mathematics)  15:00 - 17:00

Title: P-adic Hodge, prisms and Galois representations

Abstract: 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.

• May 19, 2021. Junxian Li (Max Planck Institute for Mathematics)

Title: Uniform Titchmarsh divisor problems

Abstract: 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.

Tencent VooV Meeting, ID：995 338 269

• May 26, 2021. No talk. Dissertation Defense.

• June 02, 2021. Guangwei Hu (SDU)

Title: The circle method and shifted convolution sums

Abstract: 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.

• June 09, 2021. Hongbo Yin (SDU)

Title: Borcherds Product and CM Value of Lambda function

Abstract: 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.

• June 16 & 17, 2021. Zhi Qi (Zhejiang Univeristy)  Time: June 16, 16:00-17:00,  June 17, 10:00-11:00

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

Abstract: 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.

### Schedule: Fall semester 2020

• Sep. 8, 2020. Daniel El-Baz (Graz University of Technology)

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

Abstract: 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.

VooV Meeting, ID：995 338 269

• Sep. 15, 2020. Guangshi Lü (Shandong University)

Title: On Shifted Convolution Sums of Arithmetic Functions

Abstract: 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.

VooV Meeting, ID：995 338 269

• Sep. 22, 2020.  Lilu Zhao (Shandong University)

Title: Products of primes in arithmetic progressions (in Chinese)

• Sep. 29, 2020. Xianchang Meng (University of Göttingen)   Delayed

Title: Distinct distances on hyperbolic surfaces

Abstract: Erd\H{o}s 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 $C>0$.

VooV Meeting, ID：995 338 269

• Oct. 6, 2020. Shilun Wang (University of Padova)

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

Abstract: 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.

• Oct. 15, 2020.  Biao Wang (SUNY Buffalo)  10:00-11:00   (Note the special time.)

Abstract: 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.

VooV Meeting, ID：995 338 269

• Oct. 22, 2020.  Shaoyun Yi (University of South Carolina)  10:00-11:00   (Note the special time.)

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

Abstract: 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.

VooV Meeting, ID：995 338 269

• Oct. 27, 2020.  Liyang Yang  (California Institute of Technology)

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

Abstract: 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.

VooV Meeting, ID：995 338 269

• Nov. 03, 2020. Ofir Gorodetsky (Oxford)

Title: The distribution of squarefree integers in short intervals

Abstract: 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.

VooV Meeting, ID：995 338 269

• Nov. 10, 2020. Deyu Zhang (Shandong Normal University)

Title: On the index of composition of integral ideal

Abstract: 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.

• Nov. 17, 2020. Wenguang Zhai (China University of Mining and Technology-Beijing)

Title: On a generalization of the Euler totient function

Abstract: 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.

VooV Meeting, ID：995 338 269

• Nov. 24, 2020. Zhipeng Lu (University of Göttingen)

Title: Erdos distinct distances problem in hyperbolic surfaces

Abstract: 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.

VooV Meeting, ID：995 338 269

• Dec. 01, 2020Xianchang Meng (University of Göttingen)

Title: Distinct distances on hyperbolic surfaces

Abstract: Erd\H{o}s 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 $C>0$.

VooV Meeting, ID：995 338 269

• Dec. 08, 2020. Ezra Waxman  (Technische Universität Dresden)  15:00-16:00   (Note the special time.)

Title: Random Models for Artin Twin Primes

Abstract: 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).

Zoom 会议，Zoom ID：527 845 3958

• Dec. 15, 2020.  Mengdi Wang  (Shandong University)

Title: Recent developments on polynomial Szemeredi configuration

Abstract: 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.

### Schedule: Spring semester 2020

• April 27, 2020. Zeev Rudnick (Tel Aviv University) 15:00-16:00

Title: Prime lattice points in ovals.

Abstract: 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).

Delivered remotely, join Zoom meeting. Zoom ID：635 813 7757

• May 04,2020, Jie Wu (CNRS & UPEC) 15:00-17:00  (Note the special time.)

Title: Dance with prime numbers (This talk will be in Chinese. See Chinese seminar webpage for more information.)

Delivered remotely, join Zoom meeting. Zoom ID：635 813 7757 (this doesn't work now!)   790 719 5930  (temporarily)

• May 11,2020, Yingnan Wang (Shenzhen University) 15:00-15:50  (Note the time and Zoom ID change.)

Title: On the exceptional set of the generalized Ramanujan conjecture

Abstract: 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.

Delivered remotely, join Zoom meeting. Zoom ID：678 547 5442

• May 11,2020, Xuanxuan Xiao (Macau University of Science and Technology) 16:10-17:00  (Note the time and Zoom ID change.)

Title: Circle Method in Roth Problem and in Thin Subgroup of $SL_2(\mathbb{C})$

Abstract: 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.

Delivered remotely, join Zoom meeting. Zoom ID：678 547 5442

• May 18, 2020, Ping Xi (Xi'an Jiaotong University) 15:00-16:00  (Note that we change to VooV Meeting)

Title: A weighted Selberg sieve and applications

Abstract: 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.

VooV Meeting, ID：995 338 269

• May 18, 2020, Han Wu (Queen Mary University of London) 16:15-17:15 (Note the special time.)

Title: On Motohashi's formula

Abstract: 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.

VooV Meeting, ID：995 338 269

• May 25, 2020, Zhi Qi (Zhejiang University) 15:00-16:00

Title: Bessel functions and Beyond Endoscopy

Abstract: 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 $\mathbb{C}$.

VooV Meeting, ID：995 338 269

• June 01, 2020, Bingbing Liang (IMPAN) 10:00-11:00  (Note the special time.)

Title: Mean dimension and von Neumann-Lück rank

Abstract: 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.

VooV Meeting, ID：995 338 269

• June 08, 2020, Yangbo Ye (The University of Iowa) 9:00-10:30  (Note the time and Zoom ID change.)

Title: Techniques in Number Theory

Delivered remotely, join Zoom meeting. Zoom ID：995 2657 8586

• June 08, 2020, Dan Wang (SDU) 15:00-16:00

Title: The Selberg–Delange method in short intervals for the Dedekind zeta function

Abstract: 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.

VooV Meeting, ID：995 338 269

• June 15, 2020, Wei Zhang (SDU) 15:00-16:00

Title: On cube-free numbers of the form [n^c] over special sequences and their applications

Abstract: 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.

VooV Meeting, ID：995 338 269