Cramér’s conjecture on prime gaps

Determine whether the gaps between consecutive prime numbers satisfy p_{n+1} − p_n = O((log p_n)^2).

Background

The paper shows that if primes always occur in intervals [x, x + x^{θ'}] with θ' < 1/2, then Mills’ constant ξ_3 would be transcendental (Proposition 5.2).

Cramér’s conjecture predicts that prime gaps are at most on the order of (log p_n)^2, which would imply the needed short-interval prime occurrence, thereby yielding the transcendence of ξ_3 under this conjecture.