Robin's criterion has been written in various places in MO: define Gronwall's function $$ G(n) = \frac{\sigma(n)}{n \log \log n}. $$ In 1984, Robin showed that RH is equivalent to $$ G(n) < e^\gamma, \; \; \forall n \geq 5041.$$
Robin's adviser was Jean-Louis Nicolas. There is a new equivalence due to Nicolas, G. Caveney, and J. Sondow. Define a positive integer $N$ to be $GA1$ if $N$ is composite and $G(N) \geq G(N/p)$ for all primes $p |N.$ Let $N$ be called $GA2$ if $G(N) \geq G(aN)$ for all positive integers $a,$ where in this case we allow $N$ to be prime or composite. Then RH is equivalent to the assertion that the only number that is both $GA1$ and $GA2$ is 4. See arXiv and arXiv
I learned of this because Sondow wrote to me asking for a pdf of Robin 1984. And I wrote back. So thereWhich people ought to do.
