Prime gaps distribution It is well-known that gaps between successive primes have i.e. multimodal distribution
(with peaks at $6 k$):

I'm interested to know: what is the most suitable approximation for
such weird distributions? The envelope of peaks looks like $\chi^2$  ??
Asymptotically it depends on the radical of k: basically take
$$
f(k)=\prod_{p|k,\ p&gt;2}\frac{p-1}{p-2}
$$
and compare for different k. So f(6) = 2 > 1 = f(8), so gaps of length 6 are asymptotically twice as common as gaps of length 8. (Obviously k needs to be even.)

You asked (in a comment) for a smooth envelope. Using Mertens' theorem and the Prime Number Theorem it can be shown that
$$
f(k)=O(\log\log k)
$$
and this bound is tight in the sense that there is some $\alpha$ with $f(k)&gt;\alpha\log\log k$ for infinitely many $k$. (This can be computed without too much difficulty, if desired.)

You might notice that this does not resemble the curve you have drawn. This is the result of a number of separate factors:


The envelope necessarily ignores low points, so only 2, 6, 30, ... are relevant.
There is a discretization error such that the small members do not fit the curve very nicely. It works better for large k, say $k\ge 2\cdot3\cdot5\cdot7$. The error diminishes approximately as $O(1/\log\log k)$.
Your graph uses a small number of prime gaps. This error decreases approximately as $O(k/\log x)$ where x is the number of prime gaps used. In particular, every time you double k you need to square the number of prime gaps used to keep the error roughly constant.
EMPIRICAL RESULTS:

We are talking about primes in some interval between xbar-dx to xbar+dx where xbar is up to around 10^40, and dx is up to around sqrt(xbar)

The number of gaps G between successive primes p = 6*k+1 in this interval can be approximated by a log-linear distribution:
log( G(6*k) ) = A + B*k, for k = 1, 2, ... N small enough.

Similarly, The number of gaps G between successive primes p = 6*k-1 in this interval can be approximated by a log-linear distribution:
log( G(6*k) ) = C + D*k, for k = 1, 2, ... N small enough.

The results for (6*k+1) and (6*k-1) need to be combined to get the approximation for gaps between all the primes.