# 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 $\chi^2$ ??

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– Yury Dec 31 '12 at 15:57

Asymptotically it depends on the radical of k: basically take $$f(k)=\prod_{p|k,\ p>2}\frac{p-1}{p-2}$$ 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 $f(k)>\alpha\log\log k$ for infinitely many $k$. (This can be computed without too much difficulty, if desired.)
2. 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 $O(1/\log\log k)$.
3. 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.