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

TL;DR

This paper establishes sub-Weyl bounds for twisted short character sums involving Fourier coefficients of Hecke eigenforms, advancing the understanding of their magnitude in specific ranges.

Contribution

It introduces new sub-Weyl strength bounds for twisted sums of Fourier coefficients and primitive characters, improving previous estimates in certain ranges.

Findings

Derived bounds: S(N) q N^{5/9} p^{13r/45}

Bound is non-trivial for N q (p^{r})^{2/3 - 1/60}

Applicable within specific N and p^{r} ranges

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} \leq N \leq p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\geq \left(p^{r}\right)^{\frac{2}{3}-\frac{1}{60}}$.