Speaker: Xi Ping, Xi’an Jiaotong University

Date: December 16,2016

Time:10:00-11:00 am

Location: B924, Zhixin Building, Central Campus

Sponsor: the School of Mathematics

Abstract:

For a function $F:mathbf{Z}/qmathbf{Z}rightarrow mathbf{C}$, we consider the average of $F$ over an interval $I$, where$I$ is incomplete in the sense that the length is smaller than $q$. The classical approach of Pólya-Vinogradov to short character sums is applicable only if the length of $I$ is larger than $sqrt{q}$. In this talk, we shall present how the ideas of van der Corput on analytic exponential sums can be combined with Pólya-Vinogradov to beat the $sqrt{q}$-barrier, as long as the moduli $q$ allows certain factorizations. This will lead to a method which we call ‘’arithmetic exponent pairs’’. In particular, we will focus on squarefree $q$ and $F$ as a product of certain Frobenius trace functions defined on $mathbf{F}_p$ for $pmid q$. Some applications of arithmetic exponent pairs will also be discussed. This is based on my joint work with Jie Wu.

For more information, please visit:

http://www.maths.sdu.edu.cn/when-polya-vinogradov-meets-van-der-corput.html

Edited by:Liu Huan