Prime gaps in residue classes - where CSG > 1 is possible

mart_r · 2019-12-09, 22:13

_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?:)

mart_r · 2019-12-14, 16:46

_I don't think so. But I'm probably going to need some feedback. None of the papers I've read so far discussed the topic at hand.

_While calculating the merit of a gap via g/log(p), and the Cramér-Shanks-Granville (CSG) ratio accordingly as g/log²(p), the first few values of the latter expression don't fit in the picture. Even when the right-hand bounding prime p'=p+g is used, for p=7 and p'=11, CSG=g/log²(p')=0.69566, a value that is first superseded at p=2010733. And the situation is worse for the two smaller maximal gaps. You know what I mean, we can't compare CSG ratios like that for small p.
_Recall that CSG is always M²/g, multiplied by $\varphi(q)$ when arithmetic progressions p+iq are considered. (Ah, there's the phi I was looking for!) If the calculation for CSG is altered, it's directly from the calculation for the merit M.
_For my musings, consistency is key, thus I resorted to calculating the merit M=R(p')-R(p) in terms of the formula
$R(x)=1+\sum_{n=1}^\infty \frac{log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)$
_and I was quite happy with it for some time now. The average probability of a random number x being prime, by this reasoning, is a bit smaller than 1/log(x), namely $R'(x)=\frac{1}{\log x}\sum_{n=1}^\infty \frac{\mu(n)}{n\hspace{1}x^{1-\frac{1}{n}}}$

_Yet something kept bugging me. There is another term (well, actually, two terms) in the smooth part of the famous Riemann prime counting formula, which gives a strictly increasing function for x>1 that fits perfectly between the stairs of $\pi(x)$ from the very beginning.
$Ri(x)=R(x)+\frac{\arctan\frac{\pi}{\log x}}{\pi}-\frac{1}{\log x}$
_But now [Ri(3)-Ri(2)]²/1=0.91808, a value that is only challenged by Nyman's gap with CSG=0.92064 for all primes<2⁶⁴. Things are getting more troublesome with prime gaps in arithmetic progression. The comparison in the attached table shows that it's not quite right to simply take M=Ri(p')-Ri(p). (I've just noticed that I used ln instead of log there, just don't get confused by that:)
_M=Ri(p'+½)-Ri(p+½) is good for ordinary gaps (q=2), but not for arithmetic progressions.

_The most appropriate and consistent way I could find of dealing with the measure of the gaps is to take the sum of the derivatives of Ri(x) at all integers x=p+iq for 1 $\le$ i $\le$ k where k=g/q. Ri'(x) would then serve as the probability à la Cramér.

_Cross-check: Ri(x) ~ $\sum_{i=2}^x Ri'(i)$

_Better yet: Ri(x)-Ri(c) < $\sum_{i=2}^x Ri'(i)$ < Ri(x+1)-Ri(c) where c=1.5920763885...

_What follows is that we have to distinguish the values of q mod 4.
_When q mod 4=0, $M=\sum_{i=1}^k Ri'(p+iq)$
_When q mod 4=2, $M=\sum_{i=1}^{2k} Ri'(p+\frac{iq}{2})$

_So the question goes to the reader: Is this getting out of hand?

Bobby Jacobs · 2019-12-14, 22:52

Quote:

Originally Posted by mart_r

_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)
...

Very good! Why are the CSG ratios in column 5 not always increasing?

mart_r · 2019-12-15, 10:35

Quote:

Originally Posted by Bobby Jacobs

Very good! Why are the CSG ratios in column 5 not always increasing?

The table is sorted by the absolute size of the gap, the relative size isn't necessarily always increasing. Even the merit isn't always increasing.

mart_r · 2019-12-21, 11:06

_No objections from anyone, so I have to throw in my own concerns.

_Even if Ri'(x) is as precise as possible for describing the local average probability that x is prime, I still have a few second or third thoughts about $\varphi(q)$ . Especially when q is a primorial, the actual prime count may vary measurably from the one predicted by the formula, a phenomenon I'm still trying to work out some details to (see https://www.mersenneforum.org/showthread.php?t=15250). Of course it's certainly a negligible effect for the q's I'm looking at, but it's there, and I was wondering if an error bound can be obtained.

_I'm well aware this all sounds nitpicky. But when collecting data about extraordinarily large gaps, a "fair and square" measure of the gaps should be of the essence.

_In other news, and as not even WolframAlpha could give an answer to my satisfaction (i.e. one I was hoping to find) for the series expansion at x=1, I've worked out my own

$\frac{\arctan[\frac{\pi}{\log(1+x)}]}{\pi}\hspace{2}=\hspace{2}\frac{1}{2}+\sum_{n=1}^\infty (-1)^nx^n\sum_{k=1}^{\lfloor\frac{n+1}{2}\rfloor}\frac{(-1)^{k+1}\hspace{1}[2(k-1)]!\hspace{1}s(n,2k-1)}{n!\hspace{1}\pi^{2k}}$

_where s(n,k) are Stirling numbers, and, for x> $e^\pi$ ,

$\frac{\arctan(\frac{\pi}{\log x})}{\pi}-\frac{1}{\log x}\hspace{2}=\hspace{2}\sum_{n=1}^\infty \frac{(-1)^n\hspace{1}\pi^{2n}}{(2n+1)\hspace{1}\log^{2n+1}x}$

mart_r · 2019-12-24, 16:56

Been re-reading "Prime number races" by Granville and Martin and "Cramér vs. Cramér" by Pintz again, also some of Maier's work. (The more I read it, the better I understand it.)
Conclusion: Might as well go with
$M\hspace{1}=\hspace{1}Ri(p'+\frac{q}{4})-Ri(p+\frac{q}{4})$

It's close enough to what I previously thought was the most accurate way of measuring the merit and quite easy to calculate. A trade-off, so to speak.

q=188940 / p=8356739 / g=76*q qualifies as CSG>1 even by g/log²(p')/ $\varphi(q)$

If anyone's interested, I'll post a more exhaustive list of gaps with CSG>1 when measured by the above formula.

mart_r · 2020-03-08, 15:09

Code:

  k          g          p         p'        q       r CSG p'-style by
118    3758772  144803717  148562489    31854   27287 1.0000152764 Kourbatov/Wolf 2016
179    1885228  163504573  165389801    10532    5805 1.0000704209 Kourbatov/Wolf 2016
 83   43562550    1901563   45464113   524850  327013 1.0014031914 Raab, 20.02.2020
167    2306938   82541821   84848759    13814    3171 1.0022590147 Kourbatov/Wolf 2016
118    1594416  145465687  147060103    13512    9007 1.0026889378 Kourbatov/Wolf 2016
 87   13075056    1108727   14183783   150288   56711 1.0044797434 Raab, 25.02.2020
144 1717149024    2144897 1719293921 11924646 2144897 1.0045650866 Raab, 13.02.2020: largest known p' and largest known q for an extraordinarily large gap
115    6580070    9659921   16239991    57218   47297 1.0046426332 Kourbatov/Wolf 2019
128    7044864  302145839  309190703    55038   42257 1.0048671503 Kourbatov/Wolf 2019
129    1263426   10176791   11440217     9794     825 1.0056800570 Kourbatov/Wolf 2016
135    6336090   10862323   17198413    46934   20569 1.0064940453 Kourbatov/Wolf 2019
 63     532602     355339     887941     8454     271 1.0081862161 Kourbatov/Wolf 2016
199    3108778  524646211  527754989    15622   12585 1.0098218219 Kourbatov/Wolf 2016
166    2937868   71725099   74662967    17698   12803 1.0103309882 Kourbatov/Wolf 2016
 89    3002682    8462609   11465291    33738   28109 1.0107025944 Kourbatov/Wolf 2016
135    2453760   11626561   14080321    18176   12097 1.0107626289 Kourbatov/Wolf 2016
 86   47941560   49222847   97164407   557460  166367 1.0125803645 Raab, 25.02.2020
192    5450496  366870073  372320569    28388   11949 1.0140771094 Kourbatov/Wolf 2019
 55     229350    1409633    1638983     4170     173 1.0145547849 Kourbatov/Wolf 2016
156    2823288   37906669   40729957    18098    9457 1.0162761199 Kourbatov/Wolf 2016
183    7326222  222677837  230004059    40034    8729 1.0166221904 Kourbatov/Wolf 2019
144     657504  896016139  896673643     4566    2563 1.0179389550 Kourbatov/Wolf 2016
102    5910084   51763573   57673657    57942   21367 1.0199911211 Kourbatov/Wolf 2019
211    2119706  665152001  667271707    10046    6341 1.0223668231 Kourbatov/Wolf 2016
135     411480  470669167  471080647     3048      55 1.0235488825 Kourbatov/Wolf 2019
 76   14359440    8356739   22716179   188940   43379 1.0302944159 Raab, 18.12.2019
 79     316790     726611    1043401     4010     801 1.0309808771 Kourbatov/Wolf 2016
129    2266530  198565889  200832419    17570    7319 1.0335372951 Kourbatov/Wolf 2016
115     984170    5357381    6341551     8558      73 1.0339720553 Kourbatov/Wolf 2016
115    3422630     735473    4158103    29762   21185 1.0368176014 Kourbatov/Wolf 2016
 53    2413620     355417    2769037    45540   36637 1.0386945028 Raab, 11.12.2019
104    5609136   34016537   39625673    53934   38117 1.0412524005 Kourbatov/Wolf 2019
 82    2972664    5323187    8295851    36252   30395 1.0427690852 Raab, 20.02.2020
101    4575906   20250677   24826583    45306   44201 1.0463153374 Kourbatov/Wolf 2019
147    7230930  130172279  137403209    49190   15539 1.0468373915 Kourbatov/Wolf 2019
115  132625590    2839657  135465247  1153266  533125 1.0536024200 Raab, 21.02.2020
112    1896608     164663    2061271    16934   12257 1.0598397341 Kourbatov/Wolf 2016
222    1530912  728869417  730400329     6896    3593 1.0684247390 Kourbatov/Wolf 2016
201    3415794  376981823  380397617    16994    3921 1.0703375544 Kourbatov/Wolf 2016
 78    2157480   13074917   15232397    27660   19397 1.0716522452 Kourbatov/Wolf 2019
 65     208650    3415781    3624431     3210     341 1.0786589153 Kourbatov/Wolf 2016
 81   20655000    7827217   28482217   255000  177217 1.0953885874 Raab, 19.02.2020
206    8083028  344107541  352190569    39238   29519 1.1134625422 Kourbatov/Wolf 2016
209    3906628  190071823  193978451    18692   11567 1.1480589845 Kourbatov/Wolf 2016

g = k*q
p = left-hand bounding prime
p' = right-hand bounding prime
r = p mod q
"p'-style": CSG ratio per g/phi(q)/log²(p')

I noticed Mr Kourbatov and Mr Wolf published another paper regarding prime gaps in arithmetic progression last month. Maybe I should contact them for a coordinated search?

mart_r · 2020-03-09, 19:59

g / ( $\varphi(q)$ log² p') = 1.0642...
r=83341
q=2p

Later...

Bobby Jacobs · 2020-03-22, 20:57

Interesting. What is the exact method for organizing the gaps in the first table? Why, for example, is the gap of 2 between 3 and 5 not in the list?

mart_r · 2020-03-27, 19:05

Quote:

Originally Posted by Bobby Jacobs

Interesting. What is the exact method for organizing the gaps in the first table? Why, for example, is the gap of 2 between 3 and 5 not in the list?

Recap time. In all mathematical clarity, I think

:

For each positive integer k, there is a set of values {q,p} (q: positive integer, p: prime number) such that the value for CSG is a maximum. You can think of k as the number of steps in the arithmetic progression p+q*i, where p and p'=p+q*k are prime, and for each positive integer i<k, p+q*i is composite. If either q or p is larger than a certain threshold depending on k and CSG_max, then CSG cannot be larger than CSG_max for that respective k. Furthermore, see the explanation at the end of this post. I have to elaborate a bit more, using my most recent data...

Code:

 k        gap        q         p  CSG_approx (*)
 1          6        6         5  0.3022785196
 2          4        2         7  0.4198790468
 3         18        6        43  0.4580864612
 4        144       36        13  0.5078937563
 5         80       16        17  0.4945922381
 6        216       36       181  0.5127749768
 7        420       60       491  0.5785564482
 8        320       40        89  0.6047328717
 9        558      660       509  0.5563553702
10        660       66       941  0.6217267610
11        572       52        29  0.6957819573
12       1680      140       701  0.6322530512
13       4836      372       263  0.6532943787
14       1820      130       461  0.7120451019
15        690       46       197  0.7591592880
16      18240     1140      1933  0.7397723670
17      10098      594      1213  0.7408740922
18      79380     4410      1223  0.7223904024
19       3306      174      5413  0.7408652838
20      43680     2184      7841  0.7221957974
21      80262     3822       557  0.7356639423
22     115500     5250       887  0.8308343056
23       1610       70      7151  0.8202819418
24     121680     5070      2053  0.8291269738
25       1150       46      3109  0.7597500489
26      11856      456      7283  0.9098465138
27       2214       82      1553  0.8737278189
28     329280    11760     14759  0.8621806027
29      58812     2028     12109  0.8428999967
30     110880     3696      8539  0.9673158356
31     748650    24150    225077  0.8075674871
32      12800      400      9371  0.8520823935
33     391050    11850     76607  0.8108498874
34     119952     3528     59221  0.8737815728
35      61740     1764    159737  0.8267138843
36      15984      444     35257  0.9691356428
37      20646      558     58207  0.9212072554
38    1899240    49980    146117  0.9347761396
39    1236690    31710    593689  0.8794865058
40     176800     4420      3019  0.9204205455
41     212790     5190    479023  0.8758770439
42     294336     7008     15241  0.9157336406
43     128742     2994    113209  0.8850050549
44     194568     4422     62929  1.0347442307
45     754110    16758    333857  0.9208075667
46   11408460   248010    197963  0.8559649162
47    1639830    34890    130241  0.9537642386
48    2903040    60480   1828019  0.9360724738
49      66542     1358     29669  0.9501377450
50    8389500   167790   5943139  0.9150876319
51   14372820   281820  13354567  0.8816531164
52    2717520    52260   1431047  0.9780119273
53    2413620    45540    355417  1.1428167595
54    1343952    24888    135349  0.9670359549
55     229350     4170   1409633  1.0239918543
56    1172080    20930    801337  0.9276488991
57    1393650    24450   2403677  0.9627968462
58    5614980    96810  14224709  0.9172670989
59   19866480   336720    330791  0.9322437089
60   62546400  1042440   2426279  0.9655595927
61     570228     9348   1917871  0.9276525018
62    6145440    99120  14717069  0.9901575968
63     532602     8454    355339  1.0661299147
64    3225600    50400  21226511  0.9750409914
65     208650     3210   3415781  1.0821910171
66    1216512    18432    345577  1.0553714212
67   15812670   236010    800977  1.0100437615
68     964512    14184    697979  1.0519847511
69    1820910    26390   2449313  1.0007068828
70    1016260    14518     71713  1.0299861032
71   12309270   173370   8843699  0.9783007566
72   89555760  1243830  28312943  0.9682330236
73   99430380  1362060  48296291  1.0123498941
74    4013316    54234   1929793  1.0197942618
75  126094500  1681260   3818929  1.0382605330
76   98090160  1290660   1729477  1.0909304152
77   31955154   415002   5752739  0.9847660715
78    2157480    27660  13074917  1.0809486020
79     316790     4010    726611  1.0553141458
80   17746560   221832   3144419  1.0047285893
81   20655000   255000   7827217  1.1632336984
82    2972664    36252   5323187  1.0695381429
83   43562550   524850   1901563  1.1083998142
84  117356400  1397100   1629601  1.0126819069
85   16106820   189492    270509  1.0396328686  (q>2e6 TBD)
86   47941560   557460  49222847  1.0463006323  (q>1e6 TBD)
87   13075056   150288   1108727  1.1084866852  (q>1e6 TBD)
88  130738608  1485666   7421363  1.0424690911  (q>3e6 TBD)

(*) For this table, the formula used is

$CSG_{approx}\hspace{1}=\hspace{1}\frac{[R(p+gap+\frac{q}{2})-R(p+\frac{q}{2})]^2}{gap*\varphi(q)}$

for comparative reasons. (You may notice the outcome is slightly different compared to my first table, as the formula is slightly different.) The sum of derivatives of R(x) as explained in earlier posts is to be preferred IMHO, but it slows down the searching process terribly; the approximation formula as given here is the best for this purpose. Perhaps I should make some error analysis though.

One issue I have to mention is that I only look at even values of q, since the values for the initial primes p are the same for odd q/2, except when the initial prime is 2. For 2 $\equiv$ q (mod 4), this leaves the possibility to assign q=q/2 and concurrently k=2*k. Scanning the table, we see that CSG_max at k=60 is smaller than at k=30, and at k=72 it's smaller than at k=36, this would have had consequences if q (3696 and 444 respectively) would not be divisible by 4. For example, if q was 446 at k=36, I should list the same p with q=223 at k=72. But that's not the end of the story - with odd q, the values for CSG are slightly larger. If I should also include odd values of q in my search, a couple of values in the table might be different.
(I've been thinking about this problem of making the CSG values comparable for waaay too long, somebody please stop me...)

Using the formula above, the gap between 3 and 5 has CSG=0.2956906641, where q=2 and k=1, so it's smaller in terms of CSG than the one in the table. This gap can also be represented with q=1 and k=2, in this case CSG would be 0.3127125473, which is again smaller than the value for k=2. That's way it's not listed.

mart_r · 2020-03-29, 21:54

I've taken to do a systematic search for prime gaps in arithmetic progression.

(Perhaps this thread should be renamed "Prime gaps in arithmetic progression" as I'm not keeping the data for the gaps for every residue class seperately, the amount of data generated would be enormous...)

Here I'm focussing on sequences with common difference q < ~1000 for even q. The main goal is to find a gap with CSG > 1 which is also > q², a matter considered on page 11 in https://arxiv.org/abs/2002.02115v2. This is done in agreement with the authors of the paper.

Here's the status quo:

Code:

q: common difference in arithmetic progression p+k*q
CSG: Cramér-Shanks-Granville ratio per gap/phi(q)/log(p')² where p'=p+gap
CSG_max: maximum value found for CSG up to p_limit
p: initial prime of the gap
p_limit: search limit for p'

  q  CSG_max          gap             p       p_limit
  2  0.7394648022     210      20831323    3600000000
  4  0.7678340865     684    1464395089    7200000000
  6  0.7541322181     462      39895309   10800000000
  8  0.7436946715    1424    3176384429   14400000000
 10  0.8043339165    1610    5189671259   18000000000
 12  0.8052809769    1740   12411153017   21600000000
 14  0.8096072055    2310    2955286913   25200000000
 16  0.9046153418    3728    7194688583   28800000000
 18  0.7887450235     702        194027   32400000000
 20  0.7944296249    1600       7775737   36000000000
 22  0.8617839921    2002       4160719   39600000000
 24  0.7948892517    3720   31920285041   43200000000
 26  0.7982175172    5694   38786373547   46800000000
 28  0.8452080697    3836     279251711   50400000000
 30  0.8058232068    1980      40846943   54000000000
 32  0.8965150715    3808      11911369   57600000000
 34  0.8182937153    3876      29673503   61200000000
 36  0.7872577337    5400   24166901071   64800000000
 38  0.8518980455    2280        195271   68400000000
 40  0.8172390325    6360    3785091139   72000000000
 42  0.8251402679    3780     305816297   75600000000
 44  0.7778447835    1980         77369   79200000000
 46  0.8034522392    7268     640376543   82800000000
 48  0.7973452005    7344   26302298977   86400000000
 50  0.8793539278    4950      19315399   90000000000
 52  0.7726636783    7332     432150907   93600000000
 54  0.7956494153    7344    6831679457   97200000000
 56  0.7926567144    5376      19980397  100800000000
 58  0.7831435896    8932     582284587  104400000000
 60  0.7533972457    6180    6814968631  108000000000
 62  0.7480209472   13392   40678798409  111600000000
 64  0.8613806693    7104       9370547  115200000000
 66  0.7637963086    9834  104493100477  118800000000
 68  0.7981407159   13056    6594265703  122400000000
 70  0.8282785007   11900   42252957959  126000000000
 72  0.8602785358    4824       4344187  129600000000
 74  0.9131005021    9620      26876579  133200000000
 76  0.8416067457   16796   16804480523  136800000000
 78  0.7510340136    9828   13836158729  140400000000
 80  0.7585177824   10720    1339392487  144000000000
 82  0.8163798773    2214          1553  147600000000
 84  0.8017445129   11172   29153099231  151200000000
 86  0.8236614027    9718      19002559  154800000000
 88  0.8074176319    3344         22907  158400000000
 90  0.7952351245    8460    1391789947  162000000000
 92  0.8569074437   19320    6775293947  165600000000
 94  0.7540215308   21244   55988532973  169200000000
 96  0.7835562193    6816      14461861  172800000000
 98  0.8195247347   12936     262621343  176400000000
100  0.7274868378   15500   10549382033  180000000000
102  0.7887967496   16626  139972076507  183600000000
104  0.7885883689   24648  120860909123  187200000000
106  0.7406871356   15900     666737879  190800000000
108  0.8456207760   20196  153489937273  194400000000
110  0.7577236448    8250      14618897  198000000000
112  0.8754122474   19376    2117717087  201600000000
114  0.7615636029    9348     104547589  205200000000
116  0.7729502356   26796   63916863757  208800000000
118  0.8167041620   20886    1316371417  212400000000
120  0.8057216311   14040   13628678729  216000000000
122  0.7947909820   28182   36113593211  219600000000
124  0.8316652784   30628   57483096991  223200000000
126  0.9142860299   18270   17061510127  226800000000
128  0.8323761564    7296        113623  230400000000
130  0.7938556043   21450   20138027813  234000000000
132  0.7846464407   10560      92494289  237600000000
134  0.8397671201   14740      12075073  241200000000
136  0.7672983613   26928   14785702759  244800000000
138  0.7924769327   15042    1047660137  248400000000
140  0.7831617970    8400       3096061  252000000000
142  0.8088733993   12780       3334549  255600000000
144  0.7893617069   17856    2678906773  259200000000
146  0.8196507109   23652     494696261  262800000000
148  0.7782739647   38480  240275550413  266400000000
150  0.7468321094    6300       2020723  270000000000
152  0.7468621394   28120    8536597477  273600000000
154  0.9369830338   35728   88779374809  277200000000
156  0.7985008265   13728     165640219  280800000000
158  0.7393182535   39816  258043969397  284400000000
160  0.7853708350   28960   26578052863  288000000000
162  0.8258805846   26244   34294519837  291600000000
164  0.7735979649   35260   23242155967  295200000000
166  0.8175587888   42828   94831438649  298800000000
168  0.7739486136   24864  171996125029  302400000000
170  0.7905258192   29240   27579604799  306000000000
172  0.8745407940   38356    8387732083  309600000000
174  0.8966388328   14616      25666717  313200000000
176  0.8893387228   32912    2191630061  316800000000
178  0.7718247316   44856  144798233893  320400000000
180  0.7968104593   22680   37641351263  324000000000
182  0.8820202987   27482    1082590961  327600000000
184  0.8446274883   41032   15996631921  331200000000
186  0.8021152848   15624      66824477  334800000000
188  0.7964246226   40044   14217429113  338400000000
190  0.8735201597   38950   64231886011  342000000000
192  0.8583746127   11904       2459623  345600000000
194  0.8234255916   44232   18757576997  349200000000
196  0.8703957715   44100   46355125469  352800000000
198  0.7511941018   20196    1560142237  356400000000
200  0.8255106246   39800   45870059471  360000000000
202  0.8455991259   26462      48134069  363600000000
204  0.8383878988   31212   29818321129  367200000000
206  0.8849004882   20806       3903223  370800000000
208  0.8637254761   47008   21908990083  374400000000
210  0.7536441567   13230     202008439  378000000000
212  0.8726800198   57028   76983450889  381600000000
214  0.8300985533   54570   65376711233  385200000000
216  0.8565309700   38016   60639375773  388800000000
218  0.8000713650   60386  302626514693  392400000000
220  0.8516677011   11220        363019  396000000000
222  0.7476242389   35964  168120545239  399600000000
224  0.7530876774   27552     300702751  403200000000
226  0.8205657931   65314  378168298933  406800000000
228  0.7687388396   35568  102156515693  410400000000
230  0.8278708905   49910  232943726651  414000000000
232  0.7860150605   25288      22918543  417600000000
234  0.7932876948   27144    2934877793  421200000000
236  0.8808741581   53100    7947586583  377600000000
238  0.7961571214   47838   73308073691  380800000000
240  0.7746515981   25440    6883929373  384000000000
242  0.8312839360   32912     173468021  387200000000
244  0.8153157728   64172  132595555181  390400000000
246  0.8801251829   47478  189457586089  393600000000
248  0.8449133277   22816       3249793  396800000000
250  0.9603210373   53000   15946776179  400000000000
252  0.8886298435   37800   35983220659  403200000000
254  0.7778963787   45974    2545430123  406400000000
256  0.7680769257   53248   12798115129  409600000000
258  0.7684771240   45150  305996132599  412800000000
260  0.7724350668   30160     573522601  416000000000
262  0.7939284070   51352    4866855997  314400000000
264  0.8241313527   46200  313621476739  316800000000
266  0.8801702481   39368     688831433  319200000000
268  0.7914716927   71020  210484888321  321600000000
270  0.9385992389   27000     479487689  324000000000
272  0.7985388488   59568   30497191399  326400000000
274  0.9241529945   68500   13766905759  328800000000
276  0.8452548271   33672    1738757609  331200000000
278  0.8930499724   25854       1925299  333600000000
280  0.7989596541   52360  222398285827  336000000000
282  0.8508107698   23406      32332831  338400000000
284  0.8785941526   66456   12436366727  340800000000
286  0.7858974077   62920  165094766113  343200000000
288  0.7935760257   50688  159328991309  345600000000
290  0.7487428479   55680  155143564613  348000000000
292  0.8835286349   35624      18463427  350400000000
294  0.8874698444   31458     834434033  352800000000
296  0.7407823594   59792   19144263221  355200000000
298  0.8142218681   71520   38040537103  357600000000
300  0.8694555188   31500    1746213839  360000000000
302  0.7889783882   46206     381321631  362400000000
304  0.9026153946   44688     112884059  364800000000
306  0.8374812598   13464        403553  367200000000
308  0.8043192128   57288   38073581723  369600000000
310  0.8496494922   69750  228639573389  372000000000
312  0.8318902760   51168   98393397133  374400000000
314  0.8034115353   51810     676001831  376800000000
316  0.7761217516   36024      30954479  379200000000
318  0.8218376546   39114    1952210459  381600000000
320  0.8298538635   68480  106433008811  384000000000
322  0.8573533053   54740    3560039459  386400000000
324  0.7592814852   48600   37391996273  388800000000
326  0.8083900200   80848   61763668063  391200000000
328  0.7963256514   70848   17418374759  393600000000
330  0.7638121358   28380    2288156501  396000000000
332  0.7948104095   73704   21234656767  398400000000
334  0.7853352422   91850  337046857421  400800000000
336  0.7538847566   48720  185386505651  403200000000
338  0.8637454324   43602      64873867  405600000000
340  0.8240596233   65280   63699574657  408000000000
342  0.8010075672   59850  264946609777  410400000000
344  0.9190160329   66736    1069426291  412800000000
346  0.7997930054   84078   54543153053  415200000000
348  0.7826314483   55332   81566014381  417600000000
350  0.8106891221   61950   91077187501  420000000000
352  0.7785409404   82720  155422575103  422400000000
354  0.8053164792   59826   97833706423  424800000000
356  0.8760991508  102172  151136306521  427200000000
358  0.7782235660   36516      11215003  429600000000
360  0.9190759335   23760      13367671  432000000000
362  0.7574134126   91224  171412985833  434400000000
364  0.7756310715   51324    2040279533  436800000000
366  0.8000166747   25986      13943863  439200000000
368  0.7748084867   91632  181059672907  441600000000
370  0.7987557798   81400  357568042813  444000000000
372  0.7688792740   42408    2045612389  446400000000
374  0.8532053339   58344     951272459  448800000000
376  0.8393024208   62040     506537617  451200000000
378  0.7935133452   61236  406541144927  453600000000
380  0.8062842535   70680   51924372169  456000000000
382  0.7889373294   90916   49616984707  458400000000
384  0.7939465677   43392     941956339  460800000000
386  0.7891241447   86850   24998032091  463200000000
388  0.8235266859   72556    2009863117  465600000000
390  0.7826368919   45240   45380958619  468000000000
392  0.8571708876   75656    9004407101  470400000000
394  0.8099475511   32308       1536583  472800000000
396  0.8578841605   68904  172082709217  475200000000
398  0.8791459095   89152    6737268127  477600000000
400  0.7989545605   12800          9371  480000000000
402  0.7868411869   66732  101933130811  482400000000
404  0.7941196209   83628    9237777071  484800000000
406  0.8585658511   87696   51121166533  487200000000
408  0.8204181763   66912   91749328751  408000000000
410  0.7897091793   46740     225303313  410000000000
412  0.8036159885   90228   15439142689  412000000000
414  0.8961928199   80730  221437110397  414000000000
416  0.8020117468   75296    4013157443  416000000000
418  0.7948806774  102410  415898213609  418000000000
420  0.7627980893   26880     209227433  420000000000
422  0.8690265566   57814      53634703  422000000000
424  0.7602757566   91584   28278509927  424000000000
426  0.7914098117   71994  117628436659  426000000000
428  0.8044283310   40660       5039641  428000000000
430  0.8013985116   55470     653446711  430000000000
432  0.8219377815   53568    1734713993  432000000000
434  0.7549037817   59458    1215100363  434000000000
436  0.8047788797  102896   36828863651  436000000000
438  0.7961083251   57816    5663148871  438000000000
440  0.8792757227   58520     720313157  440000000000
442  0.7596162182   99450  219117304141  442000000000
444  0.9438876646   15984         35257  444000000000
446  0.7775995907   49060      20910913  446000000000
448  0.7848311396   90944   46689332597  448000000000
450  0.7619657101   57600   79481983369  450000000000
452  0.7838385407   97632   17418654119  452000000000
454  0.7682744177  118948  232873806283  454000000000
456  0.8469687305   11856          7283  456000000000
458  0.8209257626   71448     305532509  458000000000
460  0.7705031462   72680   11329574903  460000000000
462  0.8277181464   44814    1678178797  462000000000
464  0.8354396498   55680      30933989  464000000000
466  0.8675896733   26562         70937  466000000000
468  0.7753222048   80496  458518937557  468000000000
470  0.8680891345   94000   34320032563  470000000000
472  0.8363640483   66080     103334411  472000000000
474  0.8380885352   81528   69987740731  474000000000
476  0.9122603335   97580   17812528859  476000000000
478  0.8459786783   58316      24551903  478000000000
480  0.8369451486   55200    7215375371  480000000000
482  0.8084758201   85796    1355943101  482000000000
484  0.7928372491   60984     131951387  484000000000
486  0.9670602256   66582     897783067  486000000000
488  0.8040919055  124928  112158744989  488000000000
490  0.8378578846   82320   31813190909  490000000000
492  0.7739373512   71340   26551217113  492000000000
494  0.8392228848   79040    1171149593  494000000000
496  0.7869001503   98704    8483101717  496000000000
498  0.7903333451   86652  169488659107  498000000000
500  0.7914911855  105000  153153439729  500000000000
502  0.9346253968   41164        540139  502000000000
504  0.8170280734   50400     974403323  504000000000
506  0.8272732557   71852     425671457  506000000000
508  0.8267616819  112776   12711671711  508000000000
510  0.8611973391   44880     579426871  510000000000
512  0.7850984980  121344   46897900787  512000000000
514  0.7873373864   55512      16064987  514000000000
516  0.8019834578   72756   12362133493  516000000000
518  0.8366088676  101010   18527678629  518000000000
520  0.8696185442  109200  127816392311  520000000000
522  0.7983405988   46458     120978283  522000000000
524  0.7705087224  140432  315150910859  524000000000
526  0.8115530554   94680    1460084749  526000000000
528  0.8934942692   34848       5998913  528000000000
530  0.8049360188   57240     107123117  530000000000
532  0.7920246624   30324        575077  532000000000
534  0.8161638595   88110   57019930549  534000000000
536  0.7799025504  138288  179982795691  536000000000
538  0.9016955220  112980    2457304483  538000000000
540  0.8029054239   48060     715228027  540000000000
542  0.8534940538   61246      11964913  542000000000
544  0.8392434214  126208   33574445479  544000000000
546  0.8193290227   16380        114593  546000000000
548  0.8034500145   96996    1410813787  548000000000
550  0.7885921427  109450  275839875067  550000000000
552  0.7650231302   60720    1669752961  552000000000
554  0.7914668223  147364  190546085857  554000000000
556  0.8516820096  150676   98958026977  556000000000
558  0.9544916211  106020   61428637987  558000000000
560  0.7749958441   75040    5660034091  560000000000
562  0.7812147343  151740  274490495683  337200000000
564  0.7860473296   62604    1085107883  338400000000
566  0.7891020079  127916   25853082041  339600000000
568  0.8409819101   71568      37194763  340800000000
570  0.8015346996   27930       5671207  342000000000
572  0.7872642046  100100    9913066937  343200000000
574  0.8355665622  104468    8173667093  344400000000
576  0.7897361686  103680  227188645297  345600000000
578  0.7884091107  106352    4694158901  346800000000
580  0.8716154868  135720  282082451569  348000000000
582  0.7289731501   73332    8727166157  349200000000
584  0.7923222032  106872    2504438459  350400000000
586  0.8159773780  133608   19239163373  351600000000
588  0.7761971881   58212    1499370799  352800000000
590  0.8344010904  123900   97104120731  354000000000
592  0.9093598272  152736   30757758641  355200000000
594  0.8203596194  103950  333275730829  356400000000
596  0.8752581666  116220    1578834337  357600000000
598  0.7933508876  113022   12263164867  358800000000
600  0.7956697951   19200        196303  360000000000
602  0.8671990408   55986       8882693  361200000000
604  0.8070806533  103284     932676103  362400000000
606  0.7796492138   96960   67559839063  363600000000
608  0.8125159769  141664   48492135667  364800000000
610  0.8365143363   40260       1372451  366000000000
612  0.7978540809   44064      23167169  367200000000
614  0.8275383515   68768      14281613  368400000000
616  0.7562342777  116424   99877977587  369600000000
618  0.8265107359  111240  142939098451  370800000000
620  0.7625254531   44640       6020933  372000000000
622  0.8063729000   93300     245534483  373200000000
624  0.8064991779  101088  124849011073  374400000000
626  0.8003908014  170898  229723986623  375600000000
628  0.8690691013  152604   20089808047  376800000000
630  0.7795321897   76860  231276114073  378000000000
632  0.8457792089  182648  266552475209  379200000000
634  0.8321349317  178154  201487337099  380400000000
636  0.8465112163   94128   11000930573  381600000000
638  0.7538727012   93148    1327646743  382800000000
640  0.9120077454   55040       4602251  384000000000
642  0.8152602768   77040    1475945477  385200000000
644  0.8303242949   97244    1403478863  386400000000
646  0.8447432109  135660   18004184603  387600000000
648  0.8487263784   57024      45602917  388800000000
650  0.7783532479   57850      43857397  390000000000
652  0.8380466032  132356    3876681503  391200000000
654  0.9214740453   51666       9881863  392400000000
656  0.7679775506   82000      85638121  393600000000
658  0.8049383681  151998  228935733461  394800000000
660  0.7563398502   27060       3093703  396000000000
662  0.7767298220  179402  308765222369  397200000000
664  0.8497732615  179944  108343097587  398400000000
666  0.9612073070  137196  145879569457  399600000000
668  0.8291122833  161656   33463509787  400800000000
670  0.8588802430  146730  111632773561  402000000000
672  0.8377544361   92064   24553355279  403200000000
674  0.8927385746  154346    7102999643  404400000000
676  0.7911246648   81120      74589533  405600000000
678  0.8188167676  101700   16846380763  406800000000
680  0.7847062189  129880  110374441477  408000000000
682  0.8756801286  157542   43180032209  409200000000
684  0.8371047755   63612     139850671  410400000000
686  0.8425141547  118678    3207660733  411600000000
688  0.7914089252   87376      74464177  412800000000
690  0.8298009290   93150   92923597009  414000000000
692  0.8099695701  189608  213400291319  415200000000
694  0.9352046150  222080  238517976413  416400000000
696  0.8663493208   64728      85359893  417600000000
698  0.7797477822   45370        367369  418800000000
700  0.8284455357  136500  239460526523  420000000000
702  0.7511986045   32292       1306499  421200000000
704  0.8202954082  187264  397943965597  422400000000
706  0.8695438095   40242         55201  282400000000
708  0.8977861205  144432  273087013487  283200000000
710  0.7724598204  149810  269015016161  284000000000
712  0.7796816198  153792   19083648461  284800000000
714  0.7697989836   37842       8857367  285600000000
716  0.7831859227  193320  272750735171  286400000000
718  0.7661030339  186680  214029806201  287200000000
720  0.8305548118   31680       1290463  288000000000
722  0.7617166259  131404    5674164913  288800000000
724  0.7975329263  110772     339112381  289600000000
726  0.8398626974   87120    2693610473  290400000000
728  0.9518517909  184184  180789356273  291200000000
730  0.8169767948   59130       7609781  292000000000
732  0.8289374951   34404        479797  292800000000
734  0.8603414239  198180   78573408271  293600000000
736  0.7940567828   36064         49669  294400000000
738  0.8187851952   69372     144447397  295200000000
740  0.8024558855  102120    1346388959  296000000000
742  0.8270690786   48972        912367  296800000000
744  0.7697508611  124992  197934069341  297600000000
746  0.8040409658  181278   49168511023  298400000000
748  0.7830564314  139128   17115875771  299200000000
750  0.8359892306  106500   91377216071  300000000000
752  0.7749013078  128592    1668673339  300800000000
754  0.7762539310  126672    3722989997  301600000000
756  0.7984514297   71820     728533909  302400000000
758  0.8307863368  161454    7035777923  303200000000
760  0.8587348951  151240   54915038317  304000000000
762  0.7947601914  132588  149362737379  304800000000
764  0.8072228398  182596   39444378481  305600000000
766  0.8360220240  124092     363645049  306400000000
768  0.8159361794  104448    5146249133  307200000000
770  0.8512445396   80850     435991117  308000000000
772  0.7798776397   90324      34771999  308800000000
774  0.7963577306   77400     338022701  309600000000
776  0.8694867562   95448      21931331  310400000000
778  0.7896427008  161046    9056973697  311200000000
780  0.7579427233   30420       1870951  312000000000
782  0.7925574832  168912   48563547259  312800000000
784  0.8187678710   40768        152791  313600000000
786  0.7375163313  128904  182041921063  314400000000
788  0.8598576323  115836     112344431  315200000000
790  0.7449992657  116130    5097460699  158000000000
792  0.7747793079  121176  122064575597  158400000000
794  0.8737191630  146096     839726627  158800000000
796  0.8145832820   62088        997699  159200000000
798  0.8189774650  110124   68522931313  159600000000
800  0.7854380237  124000    4429597201  160000000000
802  0.7839343019   52932        386131  160400000000
804  0.7850960183  134268  113158700561  160800000000
806  0.9079408402   82212       7638601  161200000000
808  0.7877421483  113120     169209149  161600000000
810  0.8116850129   89910    6836640043  162000000000
812  0.8851736046  157528    9883553821  162400000000
814  0.7938088485  104192     196064567  162800000000
816  0.8066643398   61200      29890303  163200000000
818  0.8335856530  112884      81576293  163600000000
820  0.8731640737   90200      63450923  164000000000
822  0.8762643516  110970    2349421741  164400000000
824  0.8675619063  149144     821531663  164800000000
826  0.7517648722  170982  126679465349  165200000000
828  0.7652805026   90252    1510362617  165600000000
830  0.9773893526  154380    3391062847  166000000000 * largest CSG value so far
832  0.8327364626  114816     169459867  166400000000
834  0.8070598355   81732     208384291  166800000000
836  0.7538505214  174724  104621735923  167200000000
838  0.8485627499   66202        791251  167600000000
840  0.8015664476   73920    3295891901  168000000000
842  0.7876682549   71570       2370917  168400000000
844  0.8376720882  181460    7295944001  168800000000
846  0.8142049032   83754     242174417  169200000000
848  0.8232748309  191648   18770276141  169600000000
850  0.9701528945   68850       2865899  170000000000
852  0.8246959945   74976      66856927  170400000000
854  0.7816720250  118706     831340553  170800000000
856  0.7892210034  221704  150881417317  171200000000
858  0.7730683037   55770      33794983  171600000000
860  0.7788782924  136740    8456940871  172000000000
862  0.7629563985  160332    3988841561  172400000000
864  0.8914155653   49248        986113  172800000000
866  0.7871098852  122972     181431827  173200000000
868  0.7913892342  123256    1079341177  173600000000
870  0.8374583079   82650    1305824189  174000000000
872  0.7561328839  190968   31683162491  174400000000
874  0.7722875504  180044   34472439539  174800000000
876  0.8255392548  135780   23908792657  175200000000
878  0.8059992615  178234    5732050237  175600000000
880  0.8181553201  134640    7058010781  176000000000
882  0.8094486988   98784    3607715213  176400000000
884  0.7567896075  177684   54800797889  176800000000
886  0.8355161599  239220  113074220731  177200000000
888  0.8258816962  101232     911045519  177600000000
890  0.8369225223  100570     105642601  178000000000

The data for all gaps >= 4*q for every q <= 890 can be found in the file attached.

2019-12-09, 22:13	#1
mart_r Dec 2008 you know...around... 3²×5×19 Posts	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=kq in terms of merit and CSG ratio for even q with smallest prime p such that p+g is also prime and p+iq 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?:)

2019-12-21, 11:06	#5
mart_r Dec 2008 you know...around... 3²·5·19 Posts	"Kein Schwein ruft mich an, keine Sau interessiert sich für mich..." _No objections from anyone, so I have to throw in my own concerns. _Even if Ri'(x) is as precise as possible for describing the local average probability that x is prime, I still have a few second or third thoughts about $\varphi(q)$ . Especially when q is a primorial, the actual prime count may vary measurably from the one predicted by the formula, a phenomenon I'm still trying to work out some details to (see https://www.mersenneforum.org/showthread.php?t=15250). Of course it's certainly a negligible effect for the q's I'm looking at, but it's there, and I was wondering if an error bound can be obtained. _I'm well aware this all sounds nitpicky. But when collecting data about extraordinarily large gaps, a "fair and square" measure of the gaps should be of the essence. _In other news, and as not even WolframAlpha could give an answer to my satisfaction (i.e. one I was hoping to find) for the series expansion at x=1, I've worked out my own $\frac{\arctan[\frac{\pi}{\log(1+x)}]}{\pi}\hspace{2}=\hspace{2}\frac{1}{2}+\sum_{n=1}^\infty (-1)^nx^n\sum_{k=1}^{\lfloor\frac{n+1}{2}\rfloor}\frac{(-1)^{k+1}\hspace{1}[2(k-1)]!\hspace{1}s(n,2k-1)}{n!\hspace{1}\pi^{2k}}$ _where s(n,k) are Stirling numbers, and, for x> $e^\pi$ , $\frac{\arctan(\frac{\pi}{\log x})}{\pi}-\frac{1}{\log x}\hspace{2}=\hspace{2}\sum_{n=1}^\infty \frac{(-1)^n\hspace{1}\pi^{2n}}{(2n+1)\hspace{1}\log^{2n+1}x}$

2020-03-09, 19:59	#8
mart_r Dec 2008 you know...around... 357₁₆ Posts	1st known example? g / ( $\varphi(q)$ log² p') = 1.0642... r=83341 q=2p Later...

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2019-12-24, 16:56	#6
mart_r Dec 2008 you know...around... 3²×5×19 Posts	Been re-reading "Prime number races" by Granville and Martin and "Cramér vs. Cramér" by Pintz again, also some of Maier's work. (The more I read it, the better I understand it.) Conclusion: Might as well go with $M\hspace{1}=\hspace{1}Ri(p'+\frac{q}{4})-Ri(p+\frac{q}{4})$ It's close enough to what I previously thought was the most accurate way of measuring the merit and quite easy to calculate. A trade-off, so to speak. q=188940 / p=8356739 / g=76*q qualifies as CSG>1 even by g/log²(p')/ $\varphi(q)$ If anyone's interested, I'll post a more exhaustive list of gaps with CSG>1 when measured by the above formula.

2020-03-22, 20:57	#9
Bobby Jacobs May 2018 281 Posts	Interesting. What is the exact method for organizing the gaps in the first table? Why, for example, is the gap of 2 between 3 and 5 not in the list?