An exponential sum involving Fourier coefficients of eigenforms for $$SL(2,\pmb {{\mathbb {Z}}})$$

Abstract

Let $$\lambda _f (n)$$ denote the normalized n-th Fourier coefficient of a holomorphic Hecke eigencuspform or a Hecke–Maass cusp form for the full modular group. In this paper we shall exhibit cancellations in the following sum: $$\begin{aligned} \sum _{N<n \leqslant 2N} \lambda _f(n) \nu (n) e\left( \alpha n^{\theta }\right) , \end{aligned}$$ where $$\alpha , \ \theta $$ are real numbers with $$0< \theta <1$$ , and $$\nu (n)$$ is either $$\mu (n)$$ or $$\Lambda (n)$$ .