Exponential sums weighted by additive functions

TL;DR

The paper develops tight bounds for exponential sums $S_f(α;X)=\sum_{n≤X} f(n) e(αn)$ when $f$ ranges over an additive class $\mathcal{F}_0$ with $f(p)=1$, yielding a bound $S_f(α;X) \ll (\frac{X}{q^{Δ}}+X^{5/6}+X^{1-Δ}q^{Δ})((\log X)^4+(\log X)F_f(X))$ for $|α-a/q|≤q^{-2}$. It then applies this result to the circle method to study the Goldbach-Vinogradov problem weighted by $Ω(n)$, proving an asymptotic for $r_Ω(N)=\sum_{n_1+n_2+n_3=N}Ω(n_1)Ω(n_2)Ω(n_3)$ with a convergent singular series $\mathfrak{S}(N,M)$ and explicit major/minor arc analysis. The work combines a Vaughan-type decomposition, Sieguel–Walfisz-type progressions for $Ω(n)$, and a detailed major/minor-arc treatment to obtain $r_Ω(N)=\mathfrak{S}(N,M)\frac{N^2}{2}+O\left(\frac{N^2(\log\log N)^3}{(\log N)^A}\right)$ as $N\to\infty$, and discusses extensions to larger additive-function classes and related generating functions. The results pave the way for weighted partition problems and potential RH-linked insights via $z^{f(n)}$-type objects and $d_z(n)$ generalizations.

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 $\sum_{n \le x} f(n) e(αn)$ where $f \in F_0$ and $e(θ) = \exp(2πi θ)$. Both $ω$, the number of distinct primes of $n$, and $Ω$, 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 $Ω$, namely we show the behavior of $r_Ω(N) = \sum_{n_1+n_2+n_3=N}Ω(n_1)Ω(n_2)Ω(n_3)$, as $N \to \infty$. Lastly, we end with a discussion of further applications of the main result.

Theorems & Definitions (23)

• Definition 1.1
• Example 1.1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:exp sum bound class F']}
• Remark 2.1
• Remark 2.2
• Lemma 3.1