### The Riemann Hypothesis Says 5040 is the Last

#### Posted by John Baez

There are many equivalent ways to phrase the Riemann Hypothesis. I just learned a charming one from this fun-filled paper:

- Jeffrey Lagarias, Euler’s constant: Euler’s work and modern developments,
*Bull. Amer. Math. Soc.***50**(2013), 527–628.

Let $\sigma(n)$ be the sum of the divisors of $n$, for example

$\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$

In 1984 Guy Robin showed that the Riemann Hypothesis is true if and only if

$\frac{\sigma(n)}{n \ln (\ln n)} \; < \; e^\gamma$

for all $n > 5040$, where $\gamma$ is Euler’s constant

$\begin{array}{ccl} \gamma &=& \displaystyle{ \lim_{n \to \infty} \left( \sum_{i = 1}^n \frac{1}{i} - \int_1^n \frac{d x}{x} \right)} \\ \\ &=& 0.5772156649015328606\dots \end{array}$

What makes this especially tantalizing is that in 1913 Gronwall showed that for any $\epsilon > 0$,

$\frac{\sigma(n)}{n \ln (\ln n)} \; < \; e^\gamma + \epsilon$

except for finitely many $n$. He did this using the Prime Number Theorem.

What’s the deal with 5040? Well, obviously, the sum of the divisors of this number is big compared to $n \ln(\ln n)$.

In fact, the only known natural numbers $n$ with

$\frac{\sigma(n)}{n \ln (\ln n)} \; \ge \; e^\gamma$

are

$3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60,$ $72, 84, 120, 180, 240, 360, 720, 840, 2520, \; and\; 5040$

(I’m leaving $1$ and $2$ off this list because $\ln(\ln n)$ is funny for these.)

*If there’s any other natural number with this property, the Riemann Hypothesis is false. If there’s not, it’s true!*

In fact, Robin showed that if the Riemann Hypothesis is false there are *infinitely many* natural numbers with this property.

On Twitter, Nicolas Tessore kindly graphed the function $\sigma(n)/(n \ln(\ln n))$ for us:

You can see 5040 there, poking its head up, looking to the right, saying “Is there anyone out there as tall as me? Or *almost* as tall as me?”

Somehow this makes the Riemann Hypothesis very vivid to me.

## Re: The Riemann Hypothesis Says 5040 is the Last

Some calculations, just for fun.

$\sigma_1(5040) = 19344$

$5040 \ln(\ln 5040) = 10800.83063291229282848974010\dots$

$\frac{\sigma_1(5040)}{5040 \ln(\ln 5040)} = 1.7909733665348811333619013505910 \dots$

$e^\gamma = 1.781072417990197985236504103107 \dots$