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.