Suppose we have $\alpha \in \mathbb{R}$. Then we know that $$ \sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \} $$ where $\| \cdot \|$ is the distance to the nearest integer. I was wondering if we put von Mangoldt function as a weight and consider the exponential sum $$ S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha), $$ can we obtain an estimate similar to above in terms of $\| \alpha \|$ and $X$?

Thank you very much!

PS Here $\Lambda$ is the von Mangoldt function, $e(m) = e^{2 \pi i m}$.

Suppose we have $\alpha \in \mathbb{R}$. Then we know that

$$\sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$

where $\| \cdot \|$ is the distance to the nearest integer. I was wondering if we put von Mangoldt function as a weight and consider the exponential sum

$$S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)$$

can we obtain an estimate similar to above in terms of $\| \alpha \|$ and $X$?

Thank you very much!

PS Here $\Lambda$ is the von Mangoldt function, $e(m) = e^{2 \pi i m}$.