Geodesic
Upper bounds for prime gaps related to Firoozbakht's conjecture
Journal of integer sequences, Tome 18 (2015) no. 11
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 $b \to 1$ as $k \to \infty $ .
Classification : 11N05
Keywords: cramér conjecture, firoozbakht conjecture, prime gap 
@article{JIS_2015__18_11_a2,
     author = {Kourbatov,  Alexei},
     title = {Upper bounds for prime gaps related to {Firoozbakht's} conjecture},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {11},
     zbl = {1390.11105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_11_a2/}
}
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%J Journal of integer sequences
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Kourbatov,  Alexei. Upper bounds for prime gaps related to Firoozbakht's conjecture. Journal of integer sequences, Tome 18 (2015) no. 11. http://geodesic.mathdoc.fr/item/JIS_2015__18_11_a2/