Upper bounds for prime gaps related to Firoozbakht's conjecture
Abstract
We study two kinds of conjectural bounds for the prime gap after the k-th prime $p_k$: (A) $p_{k+1} < (p_k)^{1+1/k}$ and (B) $p_{k+1}-p_k < \log^2 p_k - \log p_k - b$ for $k>9$. The upper bound (A) is equivalent to Firoozbakht's conjecture. We prove that (A) implies (B) with $b=1$; on the other hand, (B) with $b=1.17$ implies (A). We also give other sufficient conditions for (A) that have the form (B) with $b\to1$ as $k\to\infty$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2015
- DOI:
- 10.48550/arXiv.1506.03042
- arXiv:
- arXiv:1506.03042
- Bibcode:
- 2015arXiv150603042K
- Keywords:
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- Mathematics - Number Theory;
- 11N05
- E-Print:
- 8 pages, with Corrigendum