The below chart shows the counts of prime gaps in the number range up to $5\times10^8$.
It is clearly an interesting shape and some key features are:
- an approximate linear relationship between the log of the gap count and the gap value.
- an approximate lower bound
- an approximate upper bound
- a "thin tail" at the bottom right indicating rare large gaps
- the exception of 1 count of gap=1 (can be ignored?)
The shape seems to hold for very large number ranges, and sources/charts can be found on the internet fairly easily - eg video.
Question: What can we derive from these features of the experimental evidence?
Note: I know experimental evidence is not proof. However, experimental evidence can hint and point us towards provable insights which we might not have otherwise explored.
In particular, if anyone can explain a link between this shape and the prime number theorem would be useful. The PNT suggests the average prime gap at $x$ is $\log(x)$ but I can't seem connect that with the chart.
I would appreciate replies suitable for an audience not trained to university level mathematics.
(The chart incorrectly uses the phrase "frequency" when it should say count).
