\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation}

In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984).

The paper where he proved this is, Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées. Neuvième Série 63 (2): 187–213, ISSN 0021-7824, MR774171

Where can I find this paper? Or, any other links that shows how the inequality has been derived would be greatly appreciated.

EDIT: I will also accept the answer if anyone can outline the steps, how Robin derived his criterion.

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    $\begingroup$ It doesn't look as if this paper is available online: Elsevier only provides the issues from 1997 onwards and Gallica only those until 1994. However, every decent math library should have it, so go to the closest one nearby. Here's the MR review where you can also find a few cited works (if you have access). Here's Guy Robin's homepage. $\endgroup$ – t.b. Dec 24 '11 at 2:36
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    $\begingroup$ Well, but you could go to a university library and look from there, for example? $\endgroup$ – t.b. Dec 24 '11 at 2:50
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    $\begingroup$ My campus library has a paper copy, but they're closed for the holidays. I could send you a scan when they open in January, if you still need it then. However, you might just try e-mailing the author; I can't speak for everyone, but in my experience most academic authors are more than happy to help someone who seems genuinely interested in their work. $\endgroup$ – Ilmari Karonen Dec 24 '11 at 3:17
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    $\begingroup$ math.stackexchange.com/questions/40762/finding-a-paper $\endgroup$ – JavaMan Dec 24 '11 at 3:40
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    $\begingroup$ @IlmariKaronen I emailed the Robin pdf to Roupam Ghosh today $\endgroup$ – Will Jagy Jan 3 '12 at 21:47

Added Feb. 2018: Here is Robin (1984)

I think you would learn enough from Choie, Y.-J., Lichiardopol, N., Moree, P., and Sole, P. which can be downloaded at MAX_PLANCK

See the references, I think you would also like the Lagarias paper. Alaoglu and Erdos is available online. The general area in use here is the colossally abundant numbers, see COLOSSAL

I gave a fairly complete description of these numbers at ME

A preprint of Lagarias is on the arXiv, LAGARIAS

The C.A. numbers are in a list on the OEIS

  • $\begingroup$ Thanks for the links. Most of them I had gone through earlier. I need to check out Alaoglu and Erdos, that you mentioned. :) $\endgroup$ – Roupam Ghosh Dec 25 '11 at 4:22
  • $\begingroup$ @RoupamGhosh If RH is false, the absolutely smallest number that fails Robin's inequality need not be C.A., but there is at least one CA number that also fails the inequality. There is a nice survey by J. L. Nicolas in a book called Ramanujan Revisited. $\endgroup$ – Will Jagy Dec 25 '11 at 4:56
  • $\begingroup$ I'm still unable to find it, nor any derivation of how those inequalities imply RH (all those papers are cross-referencing, and it seems nobody reproduced the original proofs of Nicolas and Robin) $\endgroup$ – reuns Jul 10 '16 at 17:34
  • $\begingroup$ @user1952009 my websites were down for a year, recently partly back up. My email address is available in my profile, just click on my name. $\endgroup$ – Will Jagy Jul 10 '16 at 17:37
  • $\begingroup$ @user1952009 sent email with about four pdf's $\endgroup$ – Will Jagy Jul 10 '16 at 17:56

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