On triple correlation sums of Fourier coefficients of cusp forms

Fei Hou; Bin Chen; Fei Hou; Bin Chen

doi:10.3934/math.20221063

AIMS Mathematics

2022, Volume 7, Issue 10: 19359-19371. doi: 10.3934/math.20221063

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Research article

On triple correlation sums of Fourier coefficients of cusp forms

Fei Hou ^{1
,
,},
Bin Chen ²

1.
Department of Mathematics, College of Science, Xi'an University of Technology, Xi'an 710054, China
2.
Department of Mathematics, College of Mathematics and Physics, Weinan Normal University, Weinan 714099, China

Received: 17 June 2022 Revised: 02 August 2022 Accepted: 12 August 2022 Published: 01 September 2022
MSC : 11F67, 11F66

Let $ p $ be a prime. In this paper, we study the sum

$ \sum\limits_{m\ge 1} \sum\limits_{n\ge 1} a_n \lambda_g(m)\lambda_{f}(m+pn) \,U{ \left( \frac{m}{X} \right) }V{ \left ( \frac{n}{H} \right)} $

for any newforms $ g\in \mathcal{B}_k(1) $ (or $ \mathcal{B}_\lambda^\ast(1) $) and $ f\in \mathcal{B}_k(p) $ (or $ \mathcal{B}_\lambda^\ast(p) $), with the aim of determining the explicit dependence on the level, where $ {\bf{a}} = \{a_n\in\mathbb{C}\} $ is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter $ p $, and present that this type of sum is non-trivial for any given $ H, X\ge 2 $.
- automorphic forms,
- Fourier coefficients,
- triple correlation sums
Citation: Fei Hou, Bin Chen. On triple correlation sums of Fourier coefficients of cusp forms[J]. AIMS Mathematics, 2022, 7(10): 19359-19371. doi: 10.3934/math.20221063

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Abstract

Let $ p $ be a prime. In this paper, we study the sum

$ \sum\limits_{m\ge 1} \sum\limits_{n\ge 1} a_n \lambda_g(m)\lambda_{f}(m+pn) \,U{ \left( \frac{m}{X} \right) }V{ \left ( \frac{n}{H} \right)} $

for any newforms $ g\in \mathcal{B}_k(1) $ (or $ \mathcal{B}_\lambda^\ast(1) $) and $ f\in \mathcal{B}_k(p) $ (or $ \mathcal{B}_\lambda^\ast(p) $), with the aim of determining the explicit dependence on the level, where $ {\bf{a}} = \{a_n\in\mathbb{C}\} $ is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter $ p $, and present that this type of sum is non-trivial for any given $ H, X\ge 2 $.

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