Lecture on "When Pólya-Vinogradov Meets Van Der Corput"
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Lecture on "When Pólya-Vinogradov Meets Van Der Corput"
DateandTime: 2016-12-16 09:37:24

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



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.

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Edited by: Liu Huan

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