Exponential sums weighted by additive functions

TL;DR

This paper develops bounds for exponential sums involving additive functions like the number of prime factors and applies these results to analyze the Goldbach-Vinogradov problem, providing new asymptotic formulas.

Contribution

It introduces a broad class of additive functions and establishes tight bounds for their exponential sums, enabling new insights into classical additive number theory problems.

Findings

Established tight bounds for exponential sums of additive functions.

Derived asymptotic formulas for the Goldbach-Vinogradov problem involving the total number of prime factors.

Demonstrated applications of exponential sum bounds to classical problems in additive number theory.

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 $\sum_{n \le x} f(n) e(\alpha n)$ 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.