# What are the current “records” for Robin's criterion?

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)?

• 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. – Will Jagy May 29 '18 at 15:31