Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits\_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c \ln \ln n}{\ln^\beta n}$ holds for infinitely many $n$.


Statement 2 : if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\prod \limits\_{p \leq n} \frac{p}{p-1} \geq e^\gamma \ln \theta(n)+ \frac{ c \ln \theta(n)}{\theta^\beta(n)}$ holds for infinitely many $n$.


Does Statement 1 imply Statement 2 ?!


Update : Posted On MSE



On Robin's criterion for the Riemann Hypothesis
Statement 2 was proved by Jean-Louis Nicolas: see Theorem 3 in [*Petites valeurs de la fonction d'Euler*, Journal of Number Theory, 17 (1983), 375-388](http://www.sciencedirect.com/science/article/pii/0022314X83900550). More precisely, Statement 2 follows from the bound $\varliminf x^b\log f(x)<0$ in part (c) of the quoted theorem, upon noting that $\theta(x)\asymp x$ (Chebyshev's theorem).


