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Robin's theorem says that if $$\sigma(n)<e^\gamma n\log\log n$$ holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any counterexamples, then they are colossally abundant numbers, and there are infinitely many counterexamples.

Lagarias' theorem says $$\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holding for all natural numbers $n$ is equivalent to the Riemann hypothesis, where $H_n$ is the $n$th harmonic number, the sum of the reciprocals of the first $n$ positive integers. It looks like Lagarias' inequality is sharper than Robin's, so any counterexamples to Robin's inequality must also be counterexamples to Lagarias'. Is known to be impossible for integers that aren't colossally abundant numbers or of some similar more general type to be exceptions to Lagarias' inequality?

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Lagarias wrote, ""Robin showed that, if the Riemann Hypothesis is false, then there will necessarily exist a counterexample to the inequality (1.2) that is a colossally abundant number; the same property can be established for counterexamples to (1.1). (There could potentially exist other counterexamples as well)." (1.2) is your first display, (1.1) is your second display.

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  • $\begingroup$ The question was, more specifically, whether it is known if counterexamples must be colossally abundant. $\endgroup$ – Jaycob Coleman Nov 18 '13 at 12:16
  • $\begingroup$ The answer is, more specifically, Lagarias doesn't know (or, at any rate, he didn't, when he wrote the paper). $\endgroup$ – Gerry Myerson Nov 18 '13 at 12:17
  • $\begingroup$ So you think it's unlikely that any other reference will have further information? $\endgroup$ – Jaycob Coleman Nov 18 '13 at 12:19
  • $\begingroup$ I don't know. If you have access to Math Reviews online, I'd suggest you call up the Lagarias paper, and then follow up on papers that reference it. $\endgroup$ – Gerry Myerson Nov 18 '13 at 12:22

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