I became interested in the second differences of prime numbers, and as part of my amateur investigations, I wanted simple metric that would allow me to directly compare prime number sequences across the whole spectrum of prime number scales.
The second ratio is the difference of the two gaps surrounding a prime number divided by the sum of those two gaps. I visualize it as every odd prime number being at a specific position within a 1 dimensional box defined by the next prime number and the previous prime number. The second difference is the absolute position in the box, whereas the second ratio is its relative position, normalizing the box around each prime number to 1.
To this end, I started to use a relation which I call a "Second Ratio". My main question is how would I look up the state of research on grouping primes by second ratios? Mostly, I expect there's name for this that I don't know.
After dabbling in this question for several years (including getting these accepted: https://oeis.org/A295746 & https://oeis.org/A295973 ) I would like to understand other approaches to similar questions. Some gentle tips on prerequisites would not be out of place. Thank you.
*The following is an example of a graph of the dataset, using the second ratio (between -1 and 1) of a 100000 1024-bit prime numbers. Positive second ratios are on the right side of the circle, negative on the left. The length of the radiating lines are proportional to the frequency of that sr in the dataset. Only the most common are labeled. Many low frequency sr fill in the gaps between the more common.
