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Old 2020-09-08, 15:57   #28
mart_r's Avatar
Dec 2008
you know...around...

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Default number crunching and munching...

The computation has now reached 1013 for all q<=1000.

Summary for the 13-digit range of p:
No new record CSG was found between p=1012 and 1013.
The largest CSG in that range was 0.9687994043 for q=104, ranking # 5 in largest CSG over all p<=1013.
CSGmax has increased for 157 values of q.
The arithmetic mean of CSGmax 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 Gd 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:
#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?
Attached Files
File Type: txt PGAP_records per q 2020-09-07.txt (56.8 KB, 44 views)
File Type: txt PGAP_records - improvements 13-digit-p.txt (22.2 KB, 40 views)
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