Vinogradov three prime theorem with Piatetski-Shapiro primes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2019
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| author | Sun, Yu-Chen Du, Shanshan Pan, Hao |
| author_facet | Sun, Yu-Chen Du, Shanshan Pan, Hao |
| contents | We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green's transference principle due to Matomäki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for $c_i$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_12572 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Vinogradov three prime theorem with Piatetski-Shapiro primes Sun, Yu-Chen Du, Shanshan Pan, Hao Number Theory Combinatorics We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green's transference principle due to Matomäki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for $c_i$. |
| title | Vinogradov three prime theorem with Piatetski-Shapiro primes |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/1912.12572 |