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

2more comments