Upper bounds for prime gaps related to Firoozbakht's conjecture

A Kourbatov - arXiv preprint arXiv:1506.03042, 2015 - arxiv.org
arXiv preprint arXiv:1506.03042, 2015arxiv.org
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 $.
We study two kinds of conjectural bounds for the prime gap after the k-th prime : (A) and (B) for . The upper bound (A) is equivalent to Firoozbakht's conjecture. We prove that (A) implies (B) with ; on the other hand, (B) with implies (A). We also give other sufficient conditions for (A) that have the form (B) with as .
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