On triple correlation sums of Fourier coefficients of cusp forms

Abstract

<abstract><p>Let $ p $ be a prime. In this paper, we study the sum</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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)} $\end{document} </tex-math></disp-formula></p> <p>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 $.</p></abstract>