On page 4, it says "Now by Robin criterion d(n)<0 for n large enough, yielding lim sup (n->infinity) (d(n)) <= 0". If I understand the criterion from page 1 rig

On page 4, it says "Now by Robin criterion d(n)<0 for n large enough, yielding lim sup (n->infinity) (d(n)) <= 0". If I understand the criterion from page 1 right, assuming RH is false only implies that d(n) <= 0 for some n > 7! -- not for all n sufficiently large, so the limit superior is not constrained as claimed. In fact, on page 2, the paper claims "Thus, if m is bounded and n->infinity, we see that d(n)->infinity", which, if the falsity of RH did imply lim sup (n->infinity) (d(n)) <= 0, would make for an even shorter and simpler proof. Disclaimer: I've only got an undergrad in math and don't know much about the specifics of the cited papers, so I might be missing something.