Non-linear additive twist of Fourier coefficients of $GL(3) \times GL(2)$ and $GL(3)$ Maass forms

Abstract

Let $λ_π(m,n)$ be the Fourier coefficients of a Hecke-Maass cusp form $π$ for $SL(3,\mathbb{Z})$ and $λ_{f}(n)$ be the Fourier coefficients of Hecke-eigen form $f$ for $SL(2,\mathbb{Z})$. The aim of this article is to get a non-trivial bound on the sum which is non-linear additive twist of the coefficients $λ_π(m,n)$ and $λ_{f}(n)$. More precisely, for any $0 < β< 1$ and $ε>0$, we have $$\sum_{n=1}^{\infty} λ_π(r,n) \, e\left(αn^β\right) V\left(\frac{n}{X}\right) \ll_{π,ε} α\sqrtβr^{\frac{7}{6}}X^{\frac{3}{4}+\frac{9β}{28}+ ε}.$$ and $$\sum_{n=1}^{\infty} λ_π(r,n) \, λ_{f}(n) \, e\left(αn^β\right) V\left(\frac{n}{X}\right) \ll_{π, f,ε} (αβ)^{\frac{3}{2}} rX^{\frac{3}{4}+\frac{29β}{44}+ε},$$ where $V(x)$ is a smooth function supported in $[1,2]$ and satisfying $V^{(j)}(x) \ll_{j} 1$.