Minimal zero-free regions for results on primes between consecutive perfect $k$th powers
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917311407980544 |
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| author | Lee, Ethan Simpson |
| author_facet | Lee, Ethan Simpson |
| contents | We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture). In addition, we prove there is always a prime between consecutive perfect $86$th powers, and identify an integer sequence (that is a subset of the positive integers) for which there is always a prime between consecutive $70$th powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14340 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal zero-free regions for results on primes between consecutive perfect $k$th powers Lee, Ethan Simpson Number Theory 11M06, 11Y35, 11N05 We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture). In addition, we prove there is always a prime between consecutive perfect $86$th powers, and identify an integer sequence (that is a subset of the positive integers) for which there is always a prime between consecutive $70$th powers. |
| title | Minimal zero-free regions for results on primes between consecutive perfect $k$th powers |
| topic | Number Theory 11M06, 11Y35, 11N05 |
| url | https://arxiv.org/abs/2602.14340 |