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

TL;DR

This paper establishes Weyl-type bounds for twisted sums involving Fourier coefficients of modular forms and Dirichlet characters, improving the range of N for non-trivial bounds without using L-functions.

Contribution

It proves a new Weyl-type bound for twisted sums of Fourier coefficients, extending the range of N and avoiding L-function analysis.

Findings

Improved the range from N > p^{3/4} to N > p^{2/3}.

Derived bounds for S_{f,χ}(N) without L-function techniques.

Established bounds with explicit dependence on p and N.

Abstract

Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $\lambda_{f}(n)$. Let $\chi$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $\epsilon >0$, $$\sum_{|n| \ll N}\lambda_{f}(n)\chi (n) \ll_{f,\epsilon}N^{3/4 }p^{1/6}(pN)^{\epsilon}.$$ We can see an improvement of the range $N > p^{3/4}$ to the range $N > p^{2/3}$ and we get a bound for $S_{f,\chi}(N)$ without going into the $L$-function.