Large prime gaps and probabilistic models
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866916893047128064 |
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| author | Banks, William Ford, Kevin Tao, Terence |
| author_facet | Banks, William Ford, Kevin Tao, Terence |
| contents | We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_08613 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Large prime gaps and probabilistic models Banks, William Ford, Kevin Tao, Terence Number Theory Probability 11N05, 11B83 We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set. |
| title | Large prime gaps and probabilistic models |
| topic | Number Theory Probability 11N05, 11B83 |
| url | https://arxiv.org/abs/1908.08613 |