[cperciva]

Not at all obvious. It is a very open question whether prime gaps are bounded by ~(log N)^2... AFAIK, even assuming the GRH, the best upper bound is roughly N^(1/2).

[jwilk]

Indeed: https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture

[cperciva]

Yeah, that's the conjecture I was thinking of.