Exponential sums with coefficients of the logarithmic derivative of automorphic $L$-functions and applications

Abstract

Let $\Lambda_{\pi}(n)$ denote the $n$th coefficient in the Dirichlet series expansion of the logarithmic derivative of $L(s,\pi)$ associated with an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb{Q}$. In this paper, for all $\alpha$ of irrational type 1 lying in the interval $[0,1]$, we investigate the best possible estimate for the sum $\sum _{n\le x} \Lambda_{\pi}(n)e(n\alpha)$ under a certain assumption. And we consider the metric result on the exponential sum involving automorphic $L$-functions without any assumptions. Let $\Lambda(n)$ be the von Mangoldt function. Then as an application, for $\varepsilon>0$ and all $0<\alpha<1$ in a set of full Lebesgue measure (depending on $\pi$), we obtain $\sum _{n\le x} \Lambda(n)\lambda_{\pi}(n)e(n\alpha)=O(x^{\frac{5}{6}+\varepsilon})$.