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It is known that
$\lim$ $\sup \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k}=\frac{6}{\pi^2}$,

where $\gamma$ is the Euler-Mascheroni constant and $N_k$ is the $k-th$ primorial number.

But is it true that, for sufficiently large $N_k$,

$ \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k} >\frac{6}{\pi^2}$ ?

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  • $\begingroup$ It is not clear to me what you are asking. Are you asking whether $\forall k_0$ there exists a $k \geq k_0$ such that $\dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_K}>\frac{6}{\pi^2}$? That statement is at least consistent with the proven $\lim\sup$ value (and would be non-trivial to prove or disprove). The way your posed question reads to me, it would mean "$\exists k_0$ such that $\forall k>k_0 :\dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_K}>\frac{6}{\pi^2} $, and that statement is false. $\endgroup$
    – Mark Fischler
    Commented Feb 23, 2016 at 16:30
  • $\begingroup$ Actually the $\limsup$ here is really a limit. $\endgroup$
    – GH from MO
    Commented Feb 23, 2016 at 23:36
  • $\begingroup$ math.stackexchange.com/questions/1670635/… $\endgroup$
    – Will Jagy
    Commented Feb 24, 2016 at 19:51
  • $\begingroup$ $\sigma(n) = \sum_{d|n} d$ is the sum of divisors function, editing the tittle could be a good idea $\endgroup$
    – reuns
    Commented Feb 24, 2016 at 20:12
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    $\begingroup$ @GHfromMO, thanks for working it out, I couldn't tell. A general comment, Robin was a student of Nicolas, and Robin's criterion is fairly well known as an equivalent of RH, but the criterion of Nicolas seems much less well known. $\endgroup$
    – Will Jagy
    Commented Feb 24, 2016 at 22:29

2 Answers 2

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The affirmative answer to your question is equivalent to the Riemann Hypothesis, and this was observed by Solé and Planat as a consequence of Nicolas's earlier work. I recall their argument briefly.

Assume that the Riemann Hypothesis holds. Then Nicolas's Theorem 2 (a) implies readily that $\frac{\sigma(N_k)}{N_k \log\log N_k} >\frac{6}{\pi^2}e^\gamma$ holds for all $k$. Now assume that the Riemann Hypothesis fails. Then Nicolas's Theorem 3 (c) implies readily that the inequality holds for infinitely many $k$'s, and also that the inequality fails for infinitely many $k$'s.

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  • $\begingroup$ I see, this is a different arXiv article, Choie is not involved in this one. $\endgroup$
    – Will Jagy
    Commented Feb 24, 2016 at 22:32
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    $\begingroup$ @WillJagy: Yes, and this was mentioned in the article that you showed me. $\endgroup$
    – GH from MO
    Commented Feb 24, 2016 at 23:17
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From Jean-Louis Nicolas, http://math.univ-lyon1.fr/~nicolas/ in article we find that there are infinitely many primorials $n$ such that $$ \frac{1}{e^\gamma \log \log n} > \frac{\phi(n)}{n} $$ Therefore, infinitely often, $$ \frac{1}{e^\gamma \log \log n} \frac{\sigma(n)}{n}> \frac{\phi(n)}{n} \frac{\sigma(n)}{n} = \frac{\sigma(n) \phi(n)}{n^2} > \frac{6}{\pi^2} $$ from Theorem 329 and footnote on page 267 of Hardy and Wright.

I am unable to tell whether your inequality for sufficiently large primorials is equivalent to RH. You are mixing two types of behavior, extreme behavior for primorials is the Nicolas criterion, while that for colossally abundant numbers is the Robin criterion. The inequalities in Hardy and Wright linking $\phi(n)$ and $\sigma(n)$ are not tight enough to cross the conditions.

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