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Old 2019-12-09, 22:13   #1
mart_r
 
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Dec 2008
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23·5·17 Posts
Default Prime gaps in residue classes - where CSG > 1 is possible

_Inspired by a paper of A. Kourbatov and M. Wolf (https://arxiv.org/pdf/1901.03785.pdf, brought to my attention via rudy235's post https://www.mersenneforum.org/showpo...34&postcount=3), I took a venture into the issue of gaps between primes of the same residue class mod q myself.
_One of the first ideas was to make a list, similar to Dr. Nicely's list, for each q. We only have to look at even values of q, since the list looks practically the same for e.g. q=7 and q=14.
_It would look something like this (g = gap size):
Code:
g/q  q=2  q=4  q=6  q=8 ...
  1    3    3    5    3
  2    7    5   19    7
  3   23   17   43   17
  4   89   73  283   41
  5  139   83  197   61
  6  199  113  521  311
  7  113  691 1109  137
  8 1831  197 2389  457
  9  523  383 1327  647
 10  887 1321 4363 1913
 11 1129 1553 8297  673
...
_But it's rather pointless to collect an arbitrary amount of data like that, so I thought it would be more interesting to take a sort of perpendicular approach and look for record values of merit and Cramér-Shanks-Granville (CSG) ratio in each row of the list above.
_(Don't ask me why I wrote "perpendicular" here, it's just an image that conjured up in my head for the approach and I can't seem to think of any better word for it at the moment.)
_This would give us a single list like the one for the ordinary prime gaps where record hunters can hunt for new heights in terms of merit and CSG ratio.

_A maximum value for both merit and CSG ratio can be found for each g/q \in N at certain q with prime p:
Code:
      record                record
g/q   merit      q      p   CSG ratio   q      p
  1   1.044     30      7   0.35468     6      5
  2   1.760     30     19   0.44989     4      5
  3   2.267     30     61   0.46810     6     43
  4   2.782     90     29   0.56356    36     13
  5   3.118    210    503   0.52731    16     17
  6   3.518    420    503   0.53135   240     47
  7   4.103    420    379   0.58809    66    229
  8   4.293    840    577   0.62318    40     89
  9   4.676    840   1129   0.58533    62     19
 10   5.030   1260    797   0.62602    66    941
 11   5.326   1470   1559   0.72822    52     29
 12   5.607   1890   2141   0.64058   140    701
 13   5.962   2310  21211   0.67268   372    263
 14   6.481   1050   5647   0.72290   130    461
 15   6.542   9240   7621   0.77047    46    197
 16   6.969   3150   2953   0.75173  1140   1933
 17   7.267  30030  10037   0.75225   594   1213
 18   7.630   4410   1223   0.73922  4410   1223
 19   7.534  10920  62743   0.74289   174   5413
 20   8.349   9240  24413   0.72892  2184   7841
 21   8.395  11550  62597   0.75321  3822    557
 22   9.039   5250    887   0.84885  5250    887
 23   8.969  13860  88397   0.82106    70   7151
 24   9.067   5070   2053   0.84321  5070   2053
 25   9.126 117810 100003   0.76086    46   3109
 26   9.708   9240 278459   0.91336   456   7283
 27  10.044  16170 215077   0.87722    82   1553
 28  10.329  11760  14759   0.87084 11760  14759
 29  10.351  66990 341287   0.84785  2028  12109
 30  11.239  43890 220307   0.97552  3696   8539
 31  10.720  24150 225077   0.81048 24150 225077
 32  10.647 330330 1929071  0.85442   400   9371
 33  11.739 120120 655579   0.81450 11850  76607
 34  11.541  53130 1877773  0.87622  3528  59221
 35  11.542  35490 1155923  0.82734  1764 159737
 36  12.640 131670 141587   0.97006   444  35257
 37  12.140  92400 864107   0.92188   558  58207
 38  12.884  49980 146117   0.93973 49980 146117
 39  12.669 189420 906473   0.88131 31710 593689
 40  13.387  60060 4654417  0.92989  4420   3019
...
209                         1.14919 18692 190071823 (largest CSG ratio found by Kourbatov and Wolf)
...
_Here, for consistency and because otherwise the numbers for smaller p tend to be "skewed", I used Gram's variant of Riemann's prime counting formula
R(x)=1+\sum_{n=1}^\infty \frac{log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)}

for merit M=\frac{R(p+g)-R(p)}{\phi(q)}

and CSG=\frac{M^2\hspace{1}\phi(q)}{g}
_(That phi doesn't look quite right there... building TEX expressions is tedious.)

_In English, this here is looking for large prime gaps of size g=k*q in terms of merit and CSG ratio for even q with smallest prime p such that p+g is also prime and p+i*q is composite for all 0<i<k, i \in N.


_Well, all in all, this appears to be rather contrived. Does anyone even understand what I'm doing here? (Do I even understand it anymore?:)
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