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ZETA SEMINAR —— Zoom Engaging & Tasteful Arithmetics

In principle, we will meet this biweekly seminar on Thursday or Friday afternoons, but the exact date and time might be shifted a bit.

 May 14, 2020 (Thursday) || 15:00-16:00 Bingrong Huang (Shandong University) On the Rankin-Selberg problem In this talk, I will sketch two proofs of the classical result of Rankin and Selberg on the second moment of Fourier coefficients of a GL(2) holomorphic cusp form. Then I will introduce a method to break the $3/5$ barrier.

 May 14, 2020 (Thursday) || 16:10-17:10 Yingnan Wang (Shenzhen University) On the exceptional set of the generalized Ramanujan conjecture For any prime $p$ and Hecke-Maass form $phi$ on $GL(n) (n\geq 2)$, $alpha_{\phi,1}(p),\ldots,\alpha_{\phi,n}(p)$ 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.

 May 29, 2020 (Friday) || 19:50-20:50 Yuk-Kam Lau (The University of Hong Kong) On the first negative Hecke eigenvalue of automorphic forms on $GL(2,mathbb{R})$ Around a decade ago, Matomäki showed that the size of the first negative Hecke eigenvalue of a holomorphic primitive form of weight $k$ and level $N$ is bounded by $O((k^2N)^{3/8})$. We review this problem and the method of attack. Then we explore the application of this method in the case of primitive Maass forms.

 May 29, 2020 (Friday) || 21:00-22:00 Fan Ge (College of William & Mary, USA) The index conjecture in zero-sum theory We will talk about some recent progress on the index conjecture in the theory of zero-sum sequences. We will also talk about its connection with Dedekind-type sums.

 June 12, 2020 (Friday) || 15:00-16:00 Régis de la Bretèche (Institut de Mathématiques de Jussieu-PRG, France) Large and small Gál sums and applications We will introduce Gál sums and explain how to apply large and small values in some number theory problems.

 June 12, 2020 (Friday) || 16:10-17:10 Sary Drappeau (Aix-Marseille Université, France) Large moments of the Stern sequence and Thue-Morse generating function This talk will be on recent work with S. Bettin (Genova) on asymptotic formulae for large moments of objects related to $2$-regular sequences. We'll focus in particular the Stern sequence, defined by the recurrence relations$$s(1)=1, s(2n)=s(n), s(2n+1)=s(n)+s(n+1)$$and the Thue-Morse generating function$$sum_{0\leq n < 2^N} (-1)^{t(n)} \exp(2\pi i n x),$$where $t(n)$ is the parity of the sum of digits of $n$ in base $2$.

 June 28, 2020 (Sunday) || 15:00-16:00 Wen Huang (University of Science and Technology of China) Polynomial mean complexity and Logarithmic Sarnak Conjecture We will review some recent progresses in Sarnak conjecture. In particular, we will investigate the relation between {0,1}- sequences with polynomial mean complexity and the Logarithmic Sarnak conjecture. Based on works jonitly with Wang, Wang, Xu, Ye, Zhou.

 June 28, 2020 (Sunday) || 16:10-17:10 Xiaosheng Wu (Hefei University of Technology) The fourth moment of Dirichlet $L$-functions In this talk, we will introduce our recent work on an asymptotic formula for the fourth moment of Dirichlet L-functions. We will focus on the fourth moment averaged over primitive characters to modulus $q$ and over $t\in [0,T]$. An asymptotic formula with a power savings in the error term will be given for this case.

 July 03, 2020 (Friday) || 15:00-16:00 Haiyan Zhou (Nanjing Normal University) Subset sums over Galois rings Let $1\leq k\leq n$ be a positive integer and $G$ be a finitely generated abelian group. Let $D=\{a_1, a_2, \cdots, a_n\}\subset G$, $b\in G$. Let $N(k,b)$ denote the number of nonempty subsets with length $k$ which sums to $b$, i.e.,$N(k,b)= N_D(k,b)=\#\{S\subset D|\sum_{a\in S}a=b,~|S|=k\}=frac{1}{k!}\#\{x_1+x_2+\cdots+x_k=b|x_i\in D,~distinct\}.$The decision version of the $k$-subset sum problem for $D$ is to determine if $N_D(k, b)>0$. It arises from several applications in coding theory, cryptography, graph theory and some other fields. From a mathematical point of view, we are more interested in the actual value of the solution number $N_D(k,b)$. We would like to have an explicit formula or an asymptotic formula for the solution number $N_D(k,b)$, but this is apparently too much to hope for in general. However, we believe that it should be possible to obtain an asymptotic formula or the number $N_D(k,b)$ for $k$ in a certain range if $D$ is close to a large subset of $G$ with certain algebraic structure. For example, let $\mathbb{F}_q$ be the finite field with $q=p^s$ elements, where $p$ is a prime and $s\geq 1$ is an integer, it is known that if $\mathbb{F}_{q}-D$ is a small set, there is a simple asymptotic formula for $N_D(k,b)$. We are interested in the subset sums problem over a Galois ring $R=GR(p^2,p^{2r})$. Let $R(k)=\{yin R|~y=x^k~ {\rm for~some}~x\in R\}.$ We obtain the asymptotic formula of $N_{D}(n,b)$ for $D=R(k)\setminus\{0\}=R(k)^\ast$. From which we deduce that any element in $R$ is the sum of $n$ different $k$-th power under sufficient condition.

 July 03, 2020 (Friday) || 16:10-17:10 Tuoping DU (Northwest University) Arithmetic Siegel-Weil formula on Shimura varieties In this talk, I will introduce Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, of type $O(p,2)$ and $U(p,1)$. The height pairings of these cycles are related to special values of derivatives of Siegel Eisentein series, a conjecture was given by Kudla. Firstly, I will introduce the work of Sankaran and Garcia for the compact case. Lastly, I will recall our result on modular curves.

 July 10, 2020 (Friday) || 15:00-16:00 Igor Shparlinski (The University of New South Wales) Distribution of points on modular hyperbolas We'll give a survey of various interesting results about the set of points on the modular hyperbola $xy \equiv 1 \pmod p$ for a prime $p$. These results show that this curve is not a typical curve $f(x,y) \equiv 0 \pmod p$ and also have many surprising applications to other, seemingly unrelated, areas. Some of these properties can also be extended to points satisfying the congruence $xy \equiv 1 \pmod n$ for a composite $n$, where they become even more special.

 July 10, 2020 (Friday) || 16:10-17:10 Zilong He (The University of Hong Kong) Sign changes of Fourier coefficients over split and inert primes Let $\mathfrak{f}$ be a half-integral weight cusp form with Fourier coefficients $\mathfrak{a}_{\mathfrak{f}}(n)\in\mathbb{R}$. Kohnen et al. proved that if $t$ is a fixed squarefree positive integer with $\mathfrak{a}_{\mathfrak{f}}(t)\neq 0$, then there are infinitely many sign changes in the sequence $\{\mathfrak{a}_{\mathfrak{f}}(tp^{2})\}_{p~\text{primes}}$. Motivated by their work, we study the sign change problem of the subsequence $\{\mathfrak{a}_{\mathfrak{f}}(tp^{2})\}$ with $p$ running over all split (resp. inert) primes in quadratic number fields. In particular, we find that $\mathfrak{a}_{\mathfrak{f}}(tp^{2})$ (with fixed $t$) exhibits sign changes either for split primes or for inert primes when it does not exhibit sign changes for both simultaneously. We also give an explicit condition on the existence or non-existence of sign changes of the subsequences $\{\mathfrak{a}_{\mathfrak{f}}(tp^{2})}_{p~\text{split}\}$ and $\{\mathfrak{a}_{\mathfrak{f}}(tp^{2})}_{p~\text{inert}\}$.In this talk, we will briefly review the background of sign change problem of $\mathfrak{a}_{\mathfrak{f}}(n)$ and then present our results and main proofs. We will also introduce the connection between cusp forms of half-integral weight and the theory of quadratic forms, and furthermore, illustrate an example not exhibiting sign changes via the theory of spinor genera, which is another motivation for our work.This is a joint work with Dr. Ben Kane.

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