On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 1089-1104
  • ISSN: 0011-4642

Abstract

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Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL ( 2 , ℤ ) . Denote by λ f ( n ) the n th normalized Fourier coefficient of f . We are interested in the average behaviour of the sum ∑ a 2 + b 2 ≤ x λ f j ( a 2 + b 2 ) for x ≥ 1 , where a , b ∈ ℤ and j ≥ 9 is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power L -functions and Rankin-Selberg L -functions.

How to cite

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Hua, Guodong. "On the higher power moments of cusp form coefficients over sums of two squares." Czechoslovak Mathematical Journal 72.4 (2022): 1089-1104. <http://eudml.org/doc/298935>.

@article{Hua2022,
abstract = {Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma =\{\rm SL\} (2,\mathbb \{Z\})$. Denote by $\lambda _\{f\}(n)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum \[ \sum \_\{a^\{2\} + b^\{2\}\le x\} \lambda \_\{f\}^\{j\}(a^\{2\}+b^\{2\}) \] for $x\ge 1$, where $a,b\in \mathbb \{Z\}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.},
author = {Hua, Guodong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fourier coefficient; automorphic $L$-function; Langlands program},
language = {eng},
number = {4},
pages = {1089-1104},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the higher power moments of cusp form coefficients over sums of two squares},
url = {http://eudml.org/doc/298935},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hua, Guodong
TI - On the higher power moments of cusp form coefficients over sums of two squares
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1089
EP - 1104
AB - Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma ={\rm SL} (2,\mathbb {Z})$. Denote by $\lambda _{f}(n)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum \[ \sum _{a^{2} + b^{2}\le x} \lambda _{f}^{j}(a^{2}+b^{2}) \] for $x\ge 1$, where $a,b\in \mathbb {Z}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.
LA - eng
KW - Fourier coefficient; automorphic $L$-function; Langlands program
UR - http://eudml.org/doc/298935
ER -

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