Speaker:郗平     西安交通大学 Venue: B924 Time: 2016年12月16日 10:00-11:00 Title: When Pólya-Vinogradov meets van der Corput 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.