[scythe]

It's because the formula for the maximal prime gap that was used in Lemma 4 at the bottom of page 3, sqrt(x) log(x) / 8pi, is a consequence of the Riemann hypothesis. Otherwise the term becomes x^(3/4 + epsilon) which is a lot larger and messes up the argument. I think it's nontrivially useful here, but there might still be a proof without it

[gizmo686]

From briefly skimming the paper, it looks like the only use of lemma 4 is in lemma 5, where the author explicitly states that it is possible to prove without the Riemann hypothesis, if one is willing to work with a bound of about 10^6000000000000000000.