On triple correlation sums of Fourier coefficients of cusp forms

AIMS Mathematics

article

Fei Hou¹  Bin Chen² 

[1] Department of Mathematics, College of Science, Xi'an University of Technology;Department of Mathematics, College of Mathematics and Physics, Weinan Normal University

关键词: automorphic forms;    Fourier coefficients;    triple correlation sums;   

DOI  :  10.3934/math.20221063

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

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 $.

【 授权许可】

CC BY