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Guy Robin proved that

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

and can be found following the links in this post Robin 1984

its in French.

Could somebody please outline the steps of the proof and/or give any link(s), book(s) ect. for any detailed explainations preferably with as many steps and helpful points as possible?

It appears the Theorem is shown on pp 187-188 and is based on the work of Rosser et Schoenfeld as a starting point. So I suppose the specific question is why is the inequality true for the condition shown?

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1 Answers1

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The steps are too numerous to reproduce here but this book$^1$ Chapter 7

covers the theorem plus its consequence. I have not had time to absorb it all but it does go through the background and relation to the Riemann hypothosis.

There might be other dedicated sources, I'd like to hear from anybody that has come across them and what you think of this book. Hope it helps somebody - should be accessable from Universities for free.

$^1$ Equivalents of the Riemann Hypothesis Kevin Broughan, University of Waikato, New Zealand

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    So the answer is that it follows from $\theta(x) = x + O(x^\sigma \log^2 x)$ where $\sigma = \sup \Re(\rho)$ and $\sigma$ can't be improved in the remainder. Then use this expression to estimate everything you need in $\sigma(\prod_{p\le k} p)$ – reuns Mar 24 '19 at 16:07