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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.08725 |
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| _version_ | 1866918486397157376 |
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| author | Ferreira, Luan Alberto |
| author_facet | Ferreira, Luan Alberto |
| contents | We prove that given $λ\in \mathbb{R}$ such that $0 < λ< 1$, then $π(x + x^λ) - π(x) \sim \displaystyle \frac{x^λ}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's conjecture about the existence of at least two primes between two consecutive squares. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Real exponential sums over primes and prime gaps Ferreira, Luan Alberto Number Theory Primary 11N05, Secondary 11L20 We prove that given $λ\in \mathbb{R}$ such that $0 < λ< 1$, then $π(x + x^λ) - π(x) \sim \displaystyle \frac{x^λ}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's conjecture about the existence of at least two primes between two consecutive squares. |
| title | Real exponential sums over primes and prime gaps |
| topic | Number Theory Primary 11N05, Secondary 11L20 |
| url | https://arxiv.org/abs/2307.08725 |