According to Lemma 3.1 of Lagarias's paper 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 counterexamples to $\sigma(n) >e^{H_n}\log H_n$ 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).

According to Lemma 3.1 of Lagarias's paper 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).

According to Lemma 3.1 of Lagarias's paper 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 counterexamples to $\sigma(n) >e^{H_n}\log H_n$ 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 weaker than Robin's (and, of course, than the inequality in this question).

According to Lemma 3.1 of Lagarias's paper 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 counterexamples to $\sigma(n) >e^{H_n}\log H_n$ 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).