Exponential sums weighted by additive functions

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both $\omega$, the number of distinct primes of $n$, and $\Omega$, the total number primes of $n$, are members of $F_0$. As an application of the exponential sum result, we use the Hardy-Littlewood circle method to find the asymptotics of the Goldbach-Vinogradov ternary problem associated to $\Omega$, namely we show the behavior of $r_\Omega(N) = \sum_{n_1+n_2+n_3=N}\Omega(n_1)\Omega(n_2)\Omega(n_3)$, as $N \to \infty$. Lastly, we end with a discussion of further applications of the main result.