# Clarrification Regarding the Robin Inequality

I just read a paper related to the Robin Inequality, and the abstract read:

"Abstract. Let σ(n) denote the sum of divisors function. In this paper we give a simple proof of the Robin inequality (R): σ(n) < $e^γ$n log log n, for all positive integers n ≥ 5041. The Robin inequality (R) implies Riemann Hypothesis."

So I'm confused as to what the state of the claim is? I've seen that the Robin inequality is true if and only if the RH is true. And this paper is saying that they prove the inequality. Does this mean the RH is proven? I suppose I'm not too sure what is actually being claimed here. Although I have only just learned of it.

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).

He did not prove it without assuming the Riemann Hypothesis.

Robin's original paper: http://zakuski.utsa.edu/~jagy/Robin_1984.pdf

A paper in which Robin's dependency on the Riemann Hypothesis is mentioned: http://www.mpim-bonn.mpg.de/preblob/2960

• That's what I had figured. Thanks for the link! – Ryan Goulden May 10 '18 at 5:38
• Is there an English version of that? – Ryan Goulden May 10 '18 at 5:39
• Yes I have update the link to the paper with the English version – Nilotpal Kanti Sinha May 10 '18 at 5:43
• @RyanGoulden: If my answer satisfies your need, I guess you can accept the answer :) – Nilotpal Kanti Sinha Sep 12 '18 at 4:45
• My apologies! I sometimes forget. – Ryan Goulden Sep 12 '18 at 4:54