Exponential sums weighted by additive functions

2025/02/07 by Gafni, Ayla, Robles, Nicolas · 1 citation
#11L07 #11P32 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.05298

Abstract

We introduce a general class F₀ of additive functions f such that f(p) = 1 and prove a tight bound for exponential sums of the form ∑_{n ≤ x} f(n) e(αn) where f ∈ F₀ 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 F₀. 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) = ∑_n1+n2+n3=NΩ(n₁)Ω(n₂)Ω(n₃), as N → ∞. Lastly, we end with a discussion of further applications of the main result.

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