### 学术报告141: 解析数论相关报告五则

###### 时间：2022-12-09作者：点击数:_showDynClicks("wbnews", 1747959454, 5898)

Akin to the Erd\H{o}s-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any $d\in (0,1]$, among all $n$-vertex tournaments with $d\binom{n}{3}$ many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Kr\'al' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for $d\geq 1/36$.

In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers $\ell\not\equiv 2\mod 4$, there exists some constant $c_\ell>0$ such that if $d\geq 1-c_\ell$, then the number of $\ell$-cycles is also asymptotically minimized by the same extremal examples. In doing so, we answer a question of Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector with given $p$-norm for integers $q>p>1$. Joint with Tianyun Tang.

Erdos提出如下问题，是否对每个既约剩余系a(mod q)都存在两个不超过q的素数使得其乘积与aq同余。近些年该问题获得一些进展。本报告将汇报这些新进展。

We establish lower bounds for the discrete $2k$-th moment of the derivative of the Riemann zeta function at nontrivial zeros for all $k<0$ under the Riemann hypothesis (RH) and the assumption that all zeros of $\zeta(s)$ are simple. This is joint work with L. Zhao.

The zeros of Riemann zeta function and Dirichlet L-functions were studied for more than 110 years by many mathematicians. But for Hecke L-functions, little results were given. In this talk, we will introduce the zero distribution related to L-functions for GL(2), and talk about the gaps between zeros of Hecke L-functions.

In this talk, I will introduce our result on the sums containing the fractional parts of numbers. This improves the result of Mercier and Nowak. This work is joint with Liuying Wu.