With odd $\ n>9, \sigma(n) < {11\over 16} e^{\gamma} n \log \log n$?

If that's not the case, do we know anyway some upper bound better than that given by Robin's inequality, since it has been shown that it holds for all odd numbers > 9 ( Choie, YoungJu, et al. "On Robin’s criterion for the Riemann hypothesis." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 357-372, Theorem 1.2)? The condition $\ n>9$ is not necessary, I really care only about the bound. I'd be grateful if you could give me a proof, too.

• Do you know of Robins theorem which is the same as you say for all $n$ and without the $\frac{11}{16}$ factor. This statement is shown to be equivalent to the Riemann hypothesis making this a pretty pretty pretty advanced question to just 'ask for a proof'. – Winther Sep 5 '14 at 15:08
• @Winther Do you know that it has been shown that Robin's inequality holds for all odd numbers > 9 ? – Vincenzo Oliva Sep 5 '14 at 15:09
• I guess I better add this in the details. – Vincenzo Oliva Sep 5 '14 at 15:10
• No. Such a proof would probably be both advanced and long. I think you would be better of going directly to the paper that proves this. – Winther Sep 5 '14 at 15:15
• Not really, you can check, I've just cited the paper in the details. And after all, it was fairly obvious that no odd number > 5040 would have violated RI, which is why I'm asking if a better bound is known. – Vincenzo Oliva Sep 5 '14 at 15:21