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

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    $\begingroup$ Cross-posting is discouraged. Also your title doesn't describe the problem at all. $\endgroup$ – Trevor Gunn Jul 14 '17 at 23:14
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    $\begingroup$ As Statement 1 is true, you are simply asking if Statement 2 is true. $\endgroup$ – GH from MO Jul 14 '17 at 23:15
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    $\begingroup$ If 1 implies 2, then both are true (as 1 is known to be true), so they are equivalent. My point is that the way you ask the question is somewhat silly. You should have simply asked if Statement 2 is true or not, and use Statement 1 as motivation only. $\endgroup$ – GH from MO Jul 14 '17 at 23:21
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    $\begingroup$ I've replaced the former useless title "Does Statement 1 imply Statement 2". Feel free to improve it further $\endgroup$ – YCor Jul 14 '17 at 23:35
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    $\begingroup$ I've put the papers there @GHfromMO $\endgroup$ – reuns Jul 15 '17 at 18:42

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. 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).


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