Sub-Weyl strength bounds for twisted $GL(2)$ short character sums

Abstract

Let $$S(N)=\sum _{n\sim N}^{\text{smooth}}{\lambda }_f(n)\chi (n),$$ where ${\lambda }_f(n)$'s are Fourier coefficients of Hecke-eigen form, and $\chi$ is a primitive character of conductor $p^r$. In this article we prove a sub-Weyl strength bounds for $S(N)$. Indeed, we obtain $$S(N)\ll N^{\frac{5}{9}}{p}^{\frac{13r}{45}},$$ provided that $p^{13r/20}\le N\le p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\ge {\left(p^r\right)}^{\frac{2}{3}-\frac{1}{60}}$.

First draft