Averages of exponential twists of the von Mangoldt function

Xiumin Ren, Wei Zhang

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 exponen…