Given non-rational number $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, is there any estimation for the following series? $$ F: \mathbb{N} \to \mathbb{C}, $$ $$ F(n) := \sum_{p \le n} \exp\left[2\pi i \cdot \alpha p\right], \quad p \text{ is prime}. $$
Is there any simple to evaluate function $f: \mathbb{N} \to \mathbb{C}$ (e.g. polynomial), such that $$ F(n) = f(n) + O(n)? $$
Note that $F(n)$ is obviously not easy to evaluate.
P.s. if it helps, original problem was with $\alpha = \sqrt m \not\in \mathbb{N},\ m \in \mathbb{N}$.
P.p.s I got the following clue, but I'm not good at number theory by any means:)
Vinogradov has proven that when $(a,q)=1$ and $|\alpha-a/q|\le q^{-2}$ then $$ \sum_{p\le n}e^{2\pi i\alpha p}\ll n(\log n)^2(n^{-1/2}q^{1/2}+q^{-1/2}+e^{-\frac12\sqrt{\log x}}). $$
Moreover, when $\alpha=\frac aq+\beta$, it follows from Siegel-Walfisz theorem that for fixed $A>0$ there is $$ \sum_{p\le n}e^{2\pi i\alpha p}={\mu(q)\over\varphi(q)}\sum_{2\le k\le n}{e^{2\pi i\beta k}\over\log k}+O\left(q\beta n^2\over\log^An\right). $$
Proofs of these results can be found in Vaughan's The Hardy-Littlewood method.