What is upper bound for the largest prime in a counter-example for robin's inequality ?!

Assume that $\frac{\sigma(n)}{ n \ln \ln n} > e^\gamma$ for some $n>5040$, and let $p$ be the biggest prime dividing $n$ what is the known upper bound for $p$.

I did search in known papers and found upper bound for $SA$, $CA$ , and all these classes, but i want a more general upper bound not specific for an subset of numbers.

But from my search i have a pretty good guess that $p \approx \ln n$ but could not prove it ??!

  • $\begingroup$ in Robin (1984) he shows that, if RH is false, then there is a CA number violating the inequality; alright, infinitely many. pages 204 and 205, with RH false, the sequence of CA numbers has $$ \frac{\sigma(n)}{\log \log n} = e^\gamma \left( 1 + \Omega_{\pm} (\log n)^{-b} \right) $$ He does not mention other sequences of numbers $\endgroup$ – Will Jagy Mar 3 '18 at 2:30

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