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-   -   Prime gaps in residue classes - where CSG > 1 is possible (https://www.mersenneforum.org/showthread.php?t=25015)

 mart_r 2019-12-09 22:13

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

[COLOR=White]_[/COLOR]Inspired by a paper of A. Kourbatov and M. Wolf ([URL]https://arxiv.org/pdf/1901.03785.pdf[/URL], brought to my attention via rudy235's post [URL]https://www.mersenneforum.org/showpost.php?p=526334&postcount=3[/URL]), I took a venture into the issue of gaps between primes of the same residue class mod q myself.
[COLOR=White]_[/COLOR]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.
[COLOR=White]_[/COLOR]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
...[/CODE][COLOR=White]_[/COLOR]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.
[COLOR=White]_[/COLOR](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.)
[COLOR=White]_[/COLOR]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.

[COLOR=White]_[/COLOR]A maximum value for both merit and CSG ratio can be found for each g/q [TEX]\in N[/TEX] 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)
...[/CODE][COLOR=White]_[/COLOR]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
[TEX]R(x)=1+\sum_{n=1}^\infty \frac{log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)}[/TEX]

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

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

[COLOR=White]_[/COLOR]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 [TEX]\in N[/TEX].

[COLOR=White]_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?:)[/COLOR]

 mart_r 2019-12-14 16:46

Merit and CSG via Riemann's R(x) - a matter of belief?

1 Attachment(s)
[COLOR=white]_[/COLOR]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.

[COLOR=White]_[/COLOR]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.
[COLOR=white]_[/COLOR]Recall that CSG is always M²/g, multiplied by [TEX]\varphi(q)[/TEX] 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.
[COLOR=white]_[/COLOR]For my musings, consistency is key, thus I resorted to calculating the merit M=R(p')-R(p) in terms of the formula
[TEX]R(x)=1+\sum_{n=1}^\infty \frac{log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)[/TEX]
[COLOR=white]_[/COLOR]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 [TEX]R'(x)=\frac{1}{\log x}\sum_{n=1}^\infty \frac{\mu(n)}{n\hspace{1}x^{1-\frac{1}{n}}}[/TEX]

[COLOR=white]_[/COLOR]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 [TEX]\pi(x)[/TEX] from the very beginning.
[TEX]Ri(x)=R(x)+\frac{\arctan\frac{\pi}{\log x}}{\pi}-\frac{1}{\log x}[/TEX]
[COLOR=white]_[/COLOR]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<2[SUP]64[/SUP]. 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:)
[COLOR=white]_[/COLOR]M=Ri(p'+½)-Ri(p+½) is good for ordinary gaps (q=2), but not for arithmetic progressions.

[LEFT][COLOR=white]_[/COLOR]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[TEX]\le[/TEX]i[TEX]\le[/TEX]k where k=g/q. Ri'(x) would then serve as the probability à la Cramér.
[/LEFT]
[COLOR=white]_[/COLOR]Cross-check: Ri(x) ~ [TEX]\sum_{i=2}^x Ri'(i)[/TEX]

[COLOR=white]_[/COLOR]Better yet: Ri(x)-Ri(c) < [TEX]\sum_{i=2}^x Ri'(i)[/TEX] < Ri(x+1)-Ri(c) where c=1.5920763885...

[COLOR=white]_[/COLOR]What follows is that we have to distinguish the values of q mod 4.
[COLOR=white]_[/COLOR]When q mod 4=0, [TEX]M=\sum_{i=1}^k Ri'(p+iq)[/TEX]
[COLOR=white]_[/COLOR]When q mod 4=2, [TEX]M=\sum_{i=1}^{2k} Ri'(p+\frac{iq}{2})[/TEX]
[COLOR=white]
[/COLOR]
[COLOR=white]_[/COLOR]So the question goes to the reader: Is this getting out of hand?

 Bobby Jacobs 2019-12-14 22:52

[QUOTE=mart_r;532495]
[COLOR=White]_[/COLOR]A maximum value for both merit and CSG ratio can be found for each g/q [TEX]\in N[/TEX] 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)
...[/CODE][/QUOTE]

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

 mart_r 2019-12-15 10:35

[QUOTE=Bobby Jacobs;532938]Very good! Why are the CSG ratios in column 5 not always increasing?[/QUOTE]

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

"Kein Schwein ruft mich an, keine Sau interessiert sich für mich..."

[COLOR=white]_[/COLOR]No objections from anyone, so I have to throw in my own concerns.

[COLOR=white]_[/COLOR]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 [TEX]\varphi(q)[/TEX]. 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 [URL]https://www.mersenneforum.org/showthread.php?t=15250[/URL]). 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.

[COLOR=white]_[/COLOR]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.

[COLOR=White]_[/COLOR]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

[TEX]\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}}[/TEX]

[COLOR=White]_[/COLOR]where s(n,k) are Stirling numbers, and, for x>[TEX]e^\pi[/TEX],

[TEX]\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}[/TEX]

 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
[TEX]M\hspace{1}=\hspace{1}Ri(p'+\frac{q}{4})-Ri(p+\frac{q}{4})[/TEX]

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')/[TEX]\varphi(q)[/TEX]

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
[B]144 1717149024 2144897 1719293921 11924646 2144897 1.0045650866 Raab, 13.02.2020: [/B]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')[/CODE]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

1st known example?

g / ([TEX]\varphi(q)[/TEX] 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=Bobby Jacobs;540533]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?[/QUOTE]
Recap time. In all mathematical clarity, I think:smile::

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)
[/CODE](*) For this table, the formula used is

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

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 [TEX]\equiv[/TEX] 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

1 Attachment(s)
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 [URL]https://arxiv.org/abs/2002.02115v2[/URL]. 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
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444 0.9438876646 15984 35257 444000000000
446 0.7775995907 49060 20910913 446000000000
448 0.7848311396 90944 46689332597 448000000000
450 0.7619657101 57600 79481983369 450000000000
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454 0.7682744177 118948 232873806283 454000000000
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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
[/CODE]The data for all gaps >= 4*q for every q <= 890 can be found in the file attached.

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