2
$\begingroup$

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

$\endgroup$
  • 2
    $\begingroup$ Cross-posting is discouraged. Also your title doesn't describe the problem at all. $\endgroup$ – Trevor Gunn Jul 14 '17 at 23:14
  • 3
    $\begingroup$ As Statement 1 is true, you are simply asking if Statement 2 is true. $\endgroup$ – GH from MO Jul 14 '17 at 23:15
  • 3
    $\begingroup$ 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
  • 3
    $\begingroup$ 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
  • 1
    $\begingroup$ I've put the papers there @GHfromMO $\endgroup$ – reuns Jul 15 '17 at 18:42
7
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.