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

Article Type

Research Article

Publication Title

Ramanujan Journal

Abstract

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for SL(2 , Z) with normalized Fourier coefficients λf(n). Let χ be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form Sf,χ(N):=∑n∈Zλf(n)χ(n)W(nN).We show that, for 0<θ<110, Sf,χ(N)≪f,ϵN3/4+θ/2p1/6(pN)ϵ+N1-θ(pN)ϵ,which is non-trivial if N≥ p2/3+α+ϵ whereα==4θ1-6θ. Previously, such a bound was known for N≥ p3/4+ϵ.

First Page

551

Last Page

569

DOI

https://10.1007/s11139-022-00664-3

Publication Date

10-1-2023

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