The computation has now reached 10

^{13} for all q<=1000.

**Summary for the 13-digit range of p:**

No new record CSG was found between p=10

^{12} and 10

^{13}.

The largest CSG in that range was 0.9687994043 for q=104, ranking # 5 in largest CSG over all p<=10

^{13}.

CSG

_{max} has increased for 157 values of q.

The arithmetic mean of CSG

_{max} for q<=1000 has increased from 0.8258133580 to 0.8355509072.

The next interesting thing is the observation of a quick-and-dirty cluster analysis of all the gaps with CSG > 1. For most of the gaps it looks like the number of gaps G

_{d} with CSG > 1+d is approximately \(G\cdot{e^{\frac{-d}{\omega}}}\) (BTW, thanks kruoli!) with \(\omega\) being a constant around 0.036, but for gaps with CSG > 1.17 or 1.18, the actual number of gaps drops significantly compared to the expected number (and with it the would-rather-be-constant 0.036).

This is where I came from: at the moment I have 1684 gaps with CSG > 1. Sorted by CSG, the increase in the minimum CSG of the upper half of gaps, taken repetitively, remains more or less constant up to CSG ~ 1.17, but drops significantly above that level:

Code:

#gaps CSG >
1684 1 incr.
842 1.02524 0.02524
421 1.05108 0.02584
210 1.07634 0.02526
105 1.1000 0.02366
53 1.1251 0.0251
26 1.1485 0.0234
13 1.1743 0.0258
7 1.1912 0.0169
3 1.204 0.013
2 1.22 0.016

(The above mentioned 0.036 equals ~ the increase per "cluster" divided by log(2).)

So, 50% of the gaps with CSG > 1 have CSG > 1.025, 25% have CSG > 1.051 and so on.

This, again, is made with CSG via the non-conventional Sum(Ri')-formula. Taking the conventional formula gap/phi(q)/logĀ²p, the tabulated values would fluctuate much more and are generally more "out of tune".

All a bit strange, this. Until I find a satisfying explanation, the next thing I do: analysing a list of top-100-CSG gaps between consecutive primes, checking how the 100th (99th etc.) largest CSG behaves with respect to p.

Also, after long last, a new record CSG @ p=209,348,411 / q=1,415,237:

**CSG = 1.28848055169164 (conventional: 1.2174453913778)**.

Amazing, huh? Whadd'ya think? Huh?