Functiones et Approximatio Commentarii Mathematici

Oscillations of Fourier coefficients of $GL(m)$ Hecke-Maass forms and nonlinear exponential functions at primes

Yujiao Jiang and Guangshi Lü

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Abstract

Let $F(z)$ be a Hecke-Maass form for $SL(m,\mathbb{Z})$ and $A_F(n,1, \dots, 1)$ be the coefficients of $L$-function attached to $F.$ We study the cancellation of $A_F(n,1, \dots, 1)$ for twisted with a nonlinear exponential function at primes, namely the sum \begin{equation*} \sum_{n \leq N} \Lambda (n)A_F(n,1, \dots, 1)e ( \alpha n^\theta ), \end{equation*} where $0<\theta<2/m$. We also strengthen the corresponding previous results for holomorphic cusp forms for $SL(2,\mathbb{Z}),$ and improve the estimates of Ren-Ye on the resonance of exponential sums involving Fourier coefficients of a Maass form for $SL(m,\mathbb{Z})$.

Article information

Source
Funct. Approx. Comment. Math., Volume 57, Number 2 (2017), 185-204.

Dates
First available in Project Euclid: 28 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1490688023

Digital Object Identifier
doi:10.7169/facm/1623

Mathematical Reviews number (MathSciNet)
MR3732895

Zentralblatt MATH identifier
06864171

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11L07: Estimates on exponential sums 11L20: Sums over primes

Keywords
exponential sums Fourier coefficients Hecke-Maass forms

Citation

Jiang, Yujiao; Lü, Guangshi. Oscillations of Fourier coefficients of $GL(m)$ Hecke-Maass forms and nonlinear exponential functions at primes. Funct. Approx. Comment. Math. 57 (2017), no. 2, 185--204. doi:10.7169/facm/1623. https://projecteuclid.org/euclid.facm/1490688023


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