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I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \qquad\qquad\qquad(*)$$ holds for all $n > 5040$ (where $\gamma$ is the Euler–Mascheroni constant). Since $(*)$ is somewhat "quantifiable", I am wondering:

  • Suppose there is an (obviously rather huge) $n$ that violates $(*)$. Then RH is false, so $\zeta(z)$ must have a minimal non trivial zero $z_0$ off the critical line. Taking $n_0$ to be the smallest such $n$, can anything be said about an upper (or, why not, lower) bound of $\Im(z_0)$ or of $\pm\Re(z_0-\frac12)$ in terms of $n_0$?
  • Conversely, if RH is false, so there is such an (obviously rather huge) smallest $z_0$, are there bounds in terms of $|z_0|$ for the smallest $n_0> 5040$ violating $(*)$?

It is well-known that if $n_0$ exists, it must be an extremely abundant number (see Thm. 6), whence the title.

I am aware that a priori, there is no relationship between both orders of magnitude. A priori... that's why I am asking.

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    $\begingroup$ Have you worked through Robin's paper proving the equivalence? If not, that's a good place to start.... $\endgroup$
    – Greg Martin
    Commented Aug 18, 2018 at 9:12
  • $\begingroup$ @GregMartin Yes, I have gone through the scanned version zakuski.utsa.edu/~jagy/Robin_1984.pdf. The most relevant part for the current question would be section 4 ("Comportement de $\sigma(n)$ si l'hypothèse de Riemann n'est pas vraie"), which has some interesting ideas but nothing helpful here. $\endgroup$
    – Wolfgang
    Commented Aug 18, 2018 at 9:42
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    $\begingroup$ Is it obvious that your putative least violator $n_0$ must be colossally abundant (CA)? E.g. (2014) only claims XA, where CA $\not\supset$ XA $\subset$ SA. $\endgroup$
    – Francois Ziegler
    Commented Aug 18, 2018 at 10:18
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    $\begingroup$ @FrancoisZiegler Thank you! You are right, I had kept in mind from the Wikipedia page the assertion "It is known that Robin's inequality, if it ever fails to hold, will fail for a colossally abundant number n; thus the Riemann hypothesis is in fact equivalent to Robin's inequality holding for every colossally abundant number n > 5040.". At first I had just thought about counterexamples in general, but bounds of course only make sense for extremal ones. I have now corrected the title and included a link. $\endgroup$
    – Wolfgang
    Commented Aug 18, 2018 at 10:43
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    $\begingroup$ @FelipeVoloch Sure enough. Compare XA (oeis.org/A217867) with CA (oeis.org/A004490) and SA (oeis.org/A004394). If you paste "extremely abundant numbers" into the oeis search box, you have all three on the same page. $\endgroup$
    – Wolfgang
    Commented Aug 21, 2018 at 9:15

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