中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

学术报告

报告人： 林永晓 博士 （山东大学）
题  目：Some analytic aspects of automorphic forms
时  间：2022.10.27（星期四），14:00-16:00
地  点：腾讯会议：704-8726-6705       
摘 要：The Dirichlet divisor problem asks how precise one can evaluate the partial sum of the divisor function $\tau(n)$, the number of positive divisors of the integer $n$. More generally let $\lambda_F(n)$ be the Hecke eigenvalues of a $GL_d$ automorphic form $F$. We will discuss the problem of studying sums of $\lambda_F(n)$ weighted by oscillatory exponential functions and their application in bounding partial sum of the coefficients $\lambda_F(n)$. We will explain several cases where factorization of the coefficients allows one to improving some previous works. Some results by B. Huang, Q. Sun, Z. Wang, H. Zhang and the speaker in this direction will be discussed.

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报告人： 林永晓 博士 （山东大学）
题  目：Algebraic twists of automorphic $L$-functions
时  间：2022.11.3（星期四），14:00-16:00
地  点：腾讯会议：704-8726-6705       
摘 要：Let $\lambda_F(n)$ be the coefficients of an automorphic $L$-functions $L(F,s)$. One question in number theory is the distribution of $\lambda_F(n), n\geq 1$ when they are restricted to arithmetic progressions $n\equiv a (mod q)$ with varying moduli $q$. Partially motivated by the level of distribution problem for the sequence $\lambda_F(n), n\geq 1$, we will discuss the study of sums of $\lambda_F(n)$  twisted by a trace function attached to some $\ell$-adic sheaf. This talk is based on joint works with E. Kowalski, Ph. Michel, and W. Sawin.