The number of primes in short intervals and numerical calculations for Harman's sieve
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914095401271296 |
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| author | Li, Runbo |
| author_facet | Li, Runbo |
| contents | The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^θ, x]$ for some $0.52 \leqslant θ\leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all sufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001) and gives an affirmative answer to Harman and Pintz's argument. New arithmetic information, a delicate sieve decomposition, various techniques in Harman's sieve and accurate estimates for integrals are used to good effect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04458 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The number of primes in short intervals and numerical calculations for Harman's sieve Li, Runbo Number Theory The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^θ, x]$ for some $0.52 \leqslant θ\leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all sufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001) and gives an affirmative answer to Harman and Pintz's argument. New arithmetic information, a delicate sieve decomposition, various techniques in Harman's sieve and accurate estimates for integrals are used to good effect. |
| title | The number of primes in short intervals and numerical calculations for Harman's sieve |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.04458 |