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\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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    Hey J.M. I saw your comment now. Sorry for the late reply. Its hard for me to comprehend the paper as it is in French, which I do not know. I have been reading the paper, by typing the words in Google translate. Its a tiring process. Once I am done, I will surely post an answer here.2012-01-16

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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

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    @user1952009 sent email with about four pdf's2016-07-10