Averages of exponential twists of the von Mangoldt function

In this paper, we obtain some improved results for the exponential sum $\sum_{x<n\leq 2x}\Lambda(n)e(\alpha k n^{\theta})$ with $\theta\in(0,5/12),$ where $\Lambda(n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Vinogradov \cite{Va} and Murty-Srinivas \cite{Mu}.