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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 MO too, cross-posted

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closed as off-topic by Namaste, Shailesh, Daniel W. Farlow, Leucippus, Claude Leibovici Jul 15 '17 at 6:13

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    $\begingroup$ What are your thoughts, and where is your work on addressing your question? Please post your thoughts, and your efforts in addressing your own question. $\endgroup$ – Namaste Jul 15 '17 at 0:13
  • $\begingroup$ There is now an answer to this question on MO. $\endgroup$ – Trevor Gunn Jul 15 '17 at 0:24