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

Abstract

Let ${\lambda }_{\pi }(m,n)$ 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 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, for any $0<\beta <1$ and $\epsilon >0$, we have $$\sum _{n=1}^{\infty }{\lambda }_{\pi }(r,n)e\left(\alpha n^{\beta }\right)V\left(\frac{n}{X}\right){\ll }_{\pi ,\epsilon }\alpha \sqrt{\beta }{r}^{\frac{7}{6}}{X}^{\frac{3}{4}+\frac{9\beta }{28}+\epsilon }.$$ and $$\sum _{n=1}^{\infty }{\lambda }_{\pi }(r,n){\lambda }_f(n)e\left(\alpha n^{\beta }\right)V\left(\frac{n}{X}\right){\ll }_{\pi ,f,\epsilon }(\alpha \beta {)}^{\frac{3}{2}}r{X}^{\frac{3}{4}+\frac{29\beta }{44}+\epsilon },$$ where $V(x)$ is a smooth function supported in $[1,2]$ and satisfying $V^{(j)}(x){\ll }_{j}1$.

References in corpus (1)

• Distribution of mass of holomorphic cusp forms