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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2006.09947 |
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| _version_ | 1866914101054144512 |
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| author | Chen, Haipeng Fraser, Jonathan M. |
| author_facet | Chen, Haipeng Fraser, Jonathan M. |
| contents | Let $p_n$ denote the $n$th prime, and consider the function $1/n \mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parameterised family of Cramér type estimates on the gaps between successive primes. Here the parameterisation comes from the Hölder exponent. In particular, we show that Cramér's conjecture is equivalent to the map $1/n \mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n \mapsto 1/n$ is Hölder of all orders but not Lipshitz and this is independent of Cramér's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_09947 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Hölder maps and prime gaps Chen, Haipeng Fraser, Jonathan M. Number Theory Metric Geometry 11N05, 26A16 Let $p_n$ denote the $n$th prime, and consider the function $1/n \mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parameterised family of Cramér type estimates on the gaps between successive primes. Here the parameterisation comes from the Hölder exponent. In particular, we show that Cramér's conjecture is equivalent to the map $1/n \mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n \mapsto 1/n$ is Hölder of all orders but not Lipshitz and this is independent of Cramér's conjecture. |
| title | On Hölder maps and prime gaps |
| topic | Number Theory Metric Geometry 11N05, 26A16 |
| url | https://arxiv.org/abs/2006.09947 |