General asymptotic formula of Fourier coefficients of cusp forms over sum of two squares

Let λf(n) denote the nth normalized Fourier coefficient of the primitive holomorphic cusp forms of even integral weight k≥2 for the full modular group SL(2,Z). For x≥1, we are interested in the sums ∑n≤xλf(n)ℓ and ∑a2+b2≤xλf(a2+b2)ℓ. In this paper, we are able to establish the asymptotic formulae fo...

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Bibliographic Details
Published inJournal of number theory Vol. 236; pp. 214 - 229
Main Author Xu, Chenran
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.07.2022
Subjects
Cusp forms
Fourier coefficient
L-function
Power sum
Power sum
Cusp forms
Fourier coefficient
L-function
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ISSN0022-314X
1096-1658
DOI10.1016/j.jnt.2021.07.017

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Summary:Let λf(n) denote the nth normalized Fourier coefficient of the primitive holomorphic cusp forms of even integral weight k≥2 for the full modular group SL(2,Z). For x≥1, we are interested in the sums ∑n≤xλf(n)ℓ and ∑a2+b2≤xλf(a2+b2)ℓ. In this paper, we are able to establish the asymptotic formulae for general cases of the power sum for every positive integer ℓ∈N. The special cases of our results improve previous results.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2021.07.017