Upper bounds for prime gaps related to Firoozbakht's conjecture

Alexei Kourbatov

Published 2015-06-09Version 1

We study two upper bounds for the prime gap $g = p_{k+1}-p_k$ 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).