Exponential sums weighted by additive functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915794454052864 |
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| author | Gafni, Ayla Robles, Nicolas |
| author_facet | Gafni, Ayla Robles, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential sums weighted by additive functions Gafni, Ayla Robles, Nicolas Number Theory 11L07, 11P32, 11P55 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. |
| title | Exponential sums weighted by additive functions |
| topic | Number Theory 11L07, 11P32, 11P55 |
| url | https://arxiv.org/abs/2502.05298 |