High power sums of Fourier coefficients of holomorphic cusp forms and their applications

Guangwei Hu; Huixue Lao; Huimin Pan; Guangwei Hu; Huixue Lao; Huimin Pan

doi:10.3934/math.20241227

AIMS Mathematics

2024, Volume 9, Issue 9: 25166-25183. doi: 10.3934/math.20241227

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High power sums of Fourier coefficients of holomorphic cusp forms and their applications

School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250358, China

Received: 30 June 2024 Revised: 19 August 2024 Accepted: 21 August 2024 Published: 28 August 2024
MSC : 11F30, 11F66, 11N37

Let $ \lambda_f(n) $ be the $ n $th normalized Fourier coefficient of a holomorphic cusp form $ f $ for the full modular group. In this paper, we established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $ \{\lambda_f(n)\} $ and $ \{\lambda_f(n)\lambda_g(n)\} $ in short intervals, and presented some quantitative results for the number of sign changes for $ n\leq x $.
- cusp form,
- $ L $-function,
- Fourier coefficient,
- sign change
Citation: Guangwei Hu, Huixue Lao, Huimin Pan. High power sums of Fourier coefficients of holomorphic cusp forms and their applications[J]. AIMS Mathematics, 2024, 9(9): 25166-25183. doi: 10.3934/math.20241227

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Abstract

Let $ \lambda_f(n) $ be the $ n $th normalized Fourier coefficient of a holomorphic cusp form $ f $ for the full modular group. In this paper, we established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $ \{\lambda_f(n)\} $ and $ \{\lambda_f(n)\lambda_g(n)\} $ in short intervals, and presented some quantitative results for the number of sign changes for $ n\leq x $.

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