Upper bounds for prime gaps related to Firoozbakht's conjecture

TL;DR

This paper explores bounds on prime gaps related to Firoozbakht's conjecture, establishing implications between different conjectural bounds and providing conditions that support the conjecture's validity.

Contribution

It proves that a specific prime gap bound implies Firoozbakht's conjecture and vice versa, offering new sufficient conditions for the conjecture's truth.

Findings

Bound (A) implies bound (B) with b=1

Bound (B) with b=1.17 implies bound (A)

Other conditions approximate bound (A) as k increases

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 $k\to\infty$.