Averages of exponential twists of the von Mangoldt function

Abstract

In this paper, we obtain some improved results for the exponential sum ${\sum }_{x<n\le 2x}\Lambda (n)e(\alpha k{n}^{\theta })$ 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}.