On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 1055-1064
  • ISSN: 0011-4642

Abstract

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Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ℤ ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: ∑ n ≤ x λ f ( n ) λ g ( n ) i λ h ( n ) j ≪ f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 ≤ i ≤ 2 , j ≥ 5 are any fixed positive integers.

How to cite

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Hua, Guodong. "On higher moments of Hecke eigenvalues attached to cusp forms." Czechoslovak Mathematical Journal 72.4 (2022): 1055-1064. <http://eudml.org/doc/298925>.

@article{Hua2022,
abstract = {Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_\{1\}$, $k_\{2\}$ and $k_\{3\}$ for the full modular group $\Gamma =\{\rm SL\}(2,\mathbb \{Z\})$, respectively, and let $\lambda _\{f\}(n)$, $\lambda _\{g\}(n)$ and $\lambda _\{h\}(n)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions $\lambda _\{g\}(n)$, $\lambda _\{h\}(n)$ twisted by $\lambda _\{f\}(n)$ and establish the following results: \[ \sum \_\{n\le x\}\lambda \_\{f\}(n)\lambda \_\{g\}(n)^\{i\}\lambda \_\{h\}(n)^\{j\} \ll \_\{f,g,h,\varepsilon \} x^\{1- 1/2^\{i+j\} +\varepsilon \} \] for any $\varepsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.},
author = {Hua, Guodong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hecke eigenform; Fourier coefficient; Rankin-Selberg $L$-function},
language = {eng},
number = {4},
pages = {1055-1064},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On higher moments of Hecke eigenvalues attached to cusp forms},
url = {http://eudml.org/doc/298925},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hua, Guodong
TI - On higher moments of Hecke eigenvalues attached to cusp forms
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1055
EP - 1064
AB - Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_{1}$, $k_{2}$ and $k_{3}$ for the full modular group $\Gamma ={\rm SL}(2,\mathbb {Z})$, respectively, and let $\lambda _{f}(n)$, $\lambda _{g}(n)$ and $\lambda _{h}(n)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions $\lambda _{g}(n)$, $\lambda _{h}(n)$ twisted by $\lambda _{f}(n)$ and establish the following results: \[ \sum _{n\le x}\lambda _{f}(n)\lambda _{g}(n)^{i}\lambda _{h}(n)^{j} \ll _{f,g,h,\varepsilon } x^{1- 1/2^{i+j} +\varepsilon } \] for any $\varepsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.
LA - eng
KW - Hecke eigenform; Fourier coefficient; Rankin-Selberg $L$-function
UR - http://eudml.org/doc/298925
ER -

References

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