Define the function $f:\mathbb{N}\to\mathbb{R}$ as

$$ f(n)=e^\gamma n \log \log n, $$

where $e$ is Euler's constant and $\gamma$ is the Euler-Mascheroni constant. Then Robin's criterion states that for all positive integers $n>5040$, the statement

$$ \sigma(n) < f(n), $$

where $\sigma$ denotes the divisor sum function, is equivalent to the Riemann Hypothesis.

It is then natural to study the minima of the function $g$ defined by

$$ g(n)=f(n)-\sigma(n). $$

To my surprise, after (briefly) googling the subject, I found no reference to "record integers" $n$ that locally minimize $g$. Perhaps a more intuitive way to ask the same question is by studying another function $h$ defined as the quotient $\frac{\sigma(n)}{f(n)}$. However, I found no references for such studies either.

It is obvious that colossally abundant numbers attain such minima, however I am interested in whether there exists a "list" of record integers $n$ for which $\sigma(n)$ approaches $f(n)$ the most. Is there such a list (perhaps in the OEIS)?

  • $\begingroup$ see mathoverflow.net/questions/79927/… Have you succeeded in programming the CA numbers, say, for my number $\delta$ down to some positive lower bound? People seem to have trouble getting it right. $\endgroup$ – Will Jagy May 29 '18 at 15:31

You can use T.D. Noe's data of the first 1 million superabundant numbers to do the calculation you refer to. The question as you ask it is not as interesting as it seems since the function evaluated on the SA numbers is asymptotically an increasing function on n. Within Noe's data set however I believe you want term a[999956] which not surprisingly the last colossally abundant number in the list! Generally, each successive colossally abundant number appears to be a record setter in this regard so the interesting ones might be the ones which are not. You might also be interested in this related post where I conjecture an interesting reference function h(x).


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