Cramér’s conjecture on prime gaps

Establish that the gaps between consecutive primes satisfy p_{n+1}−p_n = O((log p_n)^2), as conjectured by Cramér.

Background

The paper shows that, under a sufficiently strong bound on prime gaps, one can deduce the transcendence of Mills’ constant. In this context, Cramér’s conjecture provides the desired upper bound p_{n+1}−p_n = O((log p_n)^2).

Assuming Cramér’s conjecture, the authors note that their Proposition 5.2 would imply the transcendence of ξ₃, directly linking this classical conjecture to the arithmetic nature of Mills’ constant.