The Riemann hypothesis is equivalent to


$\forall n\geqslant2,\: \sigma(n)<H\_n+e^{H\_n}\log{H\_n}$,


where $\sigma(n)$ is the divisor sum of $n$ and $H\_n$ is the nth harmonic number.


For large $n$, $H\_n$ is small. The only $n\geqslant 2$ I have found for which $\sigma(n)>e^{H\_n}\log{H\_n}$ are 2, 3, 4, 6, 12, 24, and 60. Are any other such $n$ known?



Is any $n>60$ known to have a divisor sum greater than $e^{H_n}\log({H_n})$, where $H_n$ is the nth harmonic number?
According to Lemma 3.1 of [Lagarias's paper](https://arxiv.org/abs/math/0008177) in which he introduces his criterion for RH, one has $$ e^{H\_n} \log H\_n \geq e^{\gamma} n \log \log n$$ for any $n \geq 3$. This means that any $n$ which violates your inequality would also yield a counterexample to Robin's inequality $$ \sigma (n) < e^\gamma n \log \log n.$$ Since it is known that the truth of RH implies that $5040$ is the largest counterexample, no $n$ for which $\sigma(n) >e^{H\_n}\log H\_n$ greater than $5040$ should be found. A computer check then confirms that $n=60$ is the largest counterexample. The interest in Lagarias's criterion lies in its requirement being *more* restrictive than Robin's (and, of course, than the inequality in this question).


