Non-linear Additive Twists of $GL(3) \times GL(2)$ and GL(3) Maass Forms

Abstract

Let $\lambda _{\pi }(r,n)$ be the Fourier coefficients of a Hecke-Maass cusp form $\pi$ for $SL(3,{\mathbb {Z}})$ and $\lambda _{f}(n)$ be the Fourier coefficients of a 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 $\lambda _{\pi }(m,n)$ and $\lambda _{f}(n)$. More precisely, we have

$$\begin{aligned} \sum _{n=1}^{\infty } \lambda _{\pi }(r,n) \, e\left( \alpha n^{\beta }\right) V\left( \frac{n}{X}\right) \ll _{\pi ,\epsilon } \alpha \sqrt{\beta }r^{\frac{7}{6}}X^{\frac{3}{4}+\frac{9\beta }{28}+ \epsilon }. \end{aligned}$$

and

$$\begin{aligned} \sum _{n=1}^{\infty } \lambda _{\pi }(r,n) \, \lambda _{f}(n) \, e\left( \alpha n^{\beta }\right) V\left( \frac{n}{X}\right) \ll _{\pi , f,\epsilon } (\alpha \beta )^{\frac{3}{2}} rX^{\frac{3}{4}+\frac{29\beta }{44}+\epsilon }, \end{aligned}$$

where V(x) is a smooth function supported in [1, 2] and satisfying $V^{(j)}(x) \ll _{j} 1$.