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

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.

Final verson, appeared in The Ramanujan Journal (2022)

Cited by in corpus (1)

• Bounds for short character sums for $GL(2)\times GL(3)$ twists