Exponential sums weighted by additive functions

Abstract

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 ${\sum }_{n\le x}f(n)e(\hat{I}\pm n)$ where $f\in F_0$ and $e(\hat{I}¸)=\exp (2\ddot{I}€i\hat{I}¸)$. Both $\ddot{I}‰$, the number of distinct primes of $n$, and $\hat{I}\text{c◯}$, 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 $\hat{I}\text{c◯}$, namely we show the behavior of $r_{\hat{I}}\text{c◯}(N)={\sum }_{n_1+n_2+n_3=N}\hat{I}\text{c◯}(n_1)\hat{I}\text{c◯}(n_2)\hat{I}\text{c◯}(n_3)$, as $N\to \infty$. Lastly, we end with a discussion of further applications of the main result.

26 pages