On page 4, it says &quot;Now by Robin criterion d(n)&lt;0 for n large enough, yielding lim sup (n-&gt;infinity) (d(n)) &lt;= 0&quot;. If I understand the criterion from page 1 right, assuming RH is false only implies that d(n) &lt;= 0 for some n &gt; 7! -- not for all n sufficiently large, so the limit superior is not constrained as claimed. In fact, on page 2, the paper claims &quot;Thus, if m is bounded and n-&gt;infinity, we see that d(n)-&gt;infinity&quot;, which, if the falsity of RH did imply lim sup (n-&gt;infinity) (d(n)) &lt;= 0, would make for an even shorter and simpler proof.<p>Disclaimer: I&#x27;ve only got an undergrad in math and don&#x27;t know much about the specifics of the cited papers, so I might be missing something.