20201221, 18:30  #34 
Dec 2008
you know...around...
3^{2}·71 Posts 
Updates are tedious but necessary
My monthly tribute to the world of number theory.
Largest CSG found during the past 4.3 weeks: 1.264846947 (p=113,109,089 / q=3,745,830) Code:
max. p searched: q <= 1000: 1.908e+13 1000 < q <= 2690: 1.023e+13 2690 < q <= 1e+5: 1.770e+11 1e+5 < q <= 2e+5: 5.720e+10 2e+5 < q <= 5e+5: 4.600e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 1.20e+10 2e+6 < q <= 4e+6: 3.0e+9 (only even q are examined) g/[phi(q)*log²(p_{2})]: 0.9241119774 my underappreciated formula: 1.0251848498 g/[phi(q)*log²(p_{1})]: 2.2178622671 Finding a CSG above 2 by any other measure is, IMHO, impossible. But I have to be careful here since it's an open problem how large that value can actually be. Data might suggest that a global maximum depends on the ratio log(p)/log(q), in the sense that the largest CSG are attained when log(p)/log(q) is just a little above 1. It may be that CSG cannot be larger than, say, 1.2, if log(p)/log(q) is larger than 2 or thereabouts. All very sketchy at the moment, maybe I'll write a paper about it when the pandemic is over... 
20201227, 17:41  #35 
May 2018
2^{2}×53 Posts 

20201227, 17:59  #36 
Dec 2008
you know...around...
3^{2}×71 Posts 

20210121, 21:53  #37 
Dec 2008
you know...around...
1001111111_{2} Posts 
I won't give up, not yet
Search stats:
Code:
max. p searched: q <= 1000: 1.985e+13 1000 < q <= 2690: 1.211e+13 2690 < q <= 1e+5: 2.347e+11 1e+5 < q <= 2e+5: 6.245e+10 2e+5 < q <= 5e+5: 6.050e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 1.600e+10 2e+6 < q <= 5e+6: 3.000e+9 (only even q are examined) of which 1,689 meet the conventional criterion g/[\(\varphi\)(q) log²(p+g)] > 1. There is one new record by the conventional criterion: p = 938,688,203 q = 4,200,826 = 2×7×61×4,919 k = 239 g/[φ(q) log²(p+g)] = 1.239732926499... (unconventionally 1.2777045741...) 
20210305, 22:31  #38 
Dec 2008
you know...around...
3^{2}·71 Posts 
February was too short, no time for an update
Nothing to write home about anyway...
Code:
max. p searched: q <= 1000: 2.000e+13 1000 < q <= 2690: 1.489e+13 2690 < q <= 4566: 1.002e+12 4566 < q <= 1e+5: 2.755e+11 1e+5 < q <= 2e+5: 1.553e+11 2e+5 < q <= 5e+5: 8.300e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 2.100e+10 2e+6 < q <= 5e+6: 3.000e+9 5e+6 < q <= 1e+7: reserved (only even q are examined) 
20210408, 20:47  #39 
Dec 2008
you know...around...
639_{10} Posts 
Ich hab noch mehr Daten...
More data, dedicated to science.
While I'm at it, I also search for the largest instance per q of the least prime in an arithmetic progression p=k*q+r. The most recent work I could find on this is from Li, Pratt, and Shakan: https://arxiv.org/abs/1607.02543. They searched all q < 10^{6}. I've searched even q < 4.4*10^{6} so far, so maybe this could be of use to somebody, anybody. (There's > 40 MB of raw data, so if a special analysis is needed, feel free to ask.) 
20210413, 20:17  #40 
Dec 2008
you know...around...
1177_{8} Posts 
Jeg har mye mer data...
Fooling around with the data for the first prime in an arithmetic progression, I noticed a peculiar pattern regarding the chance that p/[\(\varphi\)(q)*log²p] > x for increasing values of x.
Similar to the definitions in the LiPrattShakan paper, fix a positive integer q and let 0<r<q such that gcd(r,q)=1, let p_{q,r} denote the smallest prime number congruent to r mod q, and let p_{q}=max_{0<r<q}(p_{q,r}). In the spirit of the CramérShanksGranville ratio, set CSG'_{q}=p_{q}/[\(\varphi\)(q)*log²p_{q}]. Li, Pratt, and Shakan work with the measure p_{q}/[\(\varphi\)(q)*log(\(\varphi\)(q))*log(q)], I'll just employ this as the LPS ratio, for comparison. Simple question: how many times is CSG' or LPS larger than a certain value? For the following table, three different clusters are considered: Cluster A: even q<=4*10^{6} (2*10^{6} values) Cluster B: even 2*10^{6}<q<=4*10^{6} (10^{6} values) Cluster C: even 3*10^{6}<q<=4*10^{6} (5*10^{5} values) Code:
CSG'> Clstr.A Clstr.B Clstr.C LPS> Clstr.A Clstr.B Clstr.C 0.40 1996627 1000000 500000 0.70 1999408 1000000 500000 0.41 1995088 1000000 500000 0.72 1999055 1000000 500000 0.42 1992609 1000000 500000 0.74 1998493 999999 499999 0.43 1988686 1000000 500000 0.76 1997413 999987 499994 0.44 1982698 999997 499999 0.78 1995016 999895 499964 0.45 1972980 999938 499988 0.80 1989661 999395 499771 0.46 1956792 999600 499911 0.82 1977830 997246 498898 0.47 1929962 997846 499402 0.84 1953466 990898 496135 0.48 1886001 991912 497385 0.86 1907725 975445 489161 0.49 1816387 975942 491430 0.88 1831562 944903 474813 0.50 1714427 943252 477919 0.90 1719430 894876 450743 0.51 1578487 889177 453937 0.92 1572026 824145 415979 0.52 1413135 812869 418167 0.94 1397497 735650 371858 0.53 1229672 719759 372964 0.96 1210737 637820 322624 0.54 1042771 618754 322976 0.98 1022744 538510 272569 0.55 864320 518285 272012 1.00 847021 444505 225041 0.56 702292 424195 223792 1.02 689066 359056 181708 0.57 561281 340760 180225 1.04 553090 286155 144757 0.58 443076 269818 143035 1.06 438807 225563 113833 0.59 346363 211428 112130 1.08 344928 175629 88314 0.60 268369 163734 86810 1.10 269623 135949 68199 0.61 206632 126025 66884 1.12 209657 104630 52301 0.62 158452 96452 51162 1.14 162771 80189 40029 0.63 120954 73441 39024 1.16 125565 61153 30475 0.64 91991 55748 29613 1.18 96885 46561 23153 0.65 69697 42174 22378 1.20 74325 35258 17445 0.66 52650 31751 16761 1.22 56975 26700 13090 0.67 39852 23966 12620 1.24 43949 20353 10037 0.68 30156 18103 9562 1.26 33727 15401 7584 0.69 22769 13603 7202 1.28 25905 11678 5710 0.70 17224 10247 5426 1.30 19890 8821 4333 0.71 13026 7662 4038 1.32 15250 6678 3261 0.72 9804 5776 3075 1.34 11735 5076 2493 0.73 7405 4359 2312 1.36 9063 3870 1898 0.74 5564 3247 1719 1.38 7002 2949 1439 0.75 4249 2489 1314 1.40 5420 2234 1086 0.76 3216 1881 979 1.42 4193 1693 813 0.77 2436 1396 718 1.44 3238 1273 593 0.78 1808 1027 526 1.46 2490 981 454 0.79 1360 777 403 1.48 1917 728 338 0.80 1020 583 302 1.50 1488 556 259 0.81 773 442 222 1.52 1148 419 193 0.82 578 326 156 1.54 876 316 137 0.83 437 241 118 1.56 670 232 99 0.84 311 164 80 1.58 509 169 74 0.85 237 125 62 1.60 397 123 56 0.86 181 89 45 1.62 313 91 42 0.87 140 71 34 1.64 249 73 32 0.88 100 53 27 1.66 191 55 26 0.89 78 39 21 1.68 148 40 19 0.90 58 28 16 1.70 118 28 13 0.91 46 21 14 1.72 89 24 13 0.92 32 15 8 1.74 71 17 11 0.93 28 14 8 1.76 57 12 8 0.94 24 12 8 1.78 43 11 7 0.95 14 7 6 1.80 32 6 5 0.96 12 5 4 1.82 28 5 4 0.97 8 2 1 1.84 24 4 3 0.98 6 2 1 1.86 20 2 1 0.99 4 1 0 1.88 16 1 0 1.00 2 1 0 1.90 15 1 0 The heuristics of Li, Pratt, and Shakan suggest, or so I suppose, that eventually half of the time either LPS<1 or LPS>1. Anyway I concentrate more on CSG' as this measure is more akin to the one used in dealing with the usual prime gaps. It's hard to compare LPS and CSG' directly (by trying to put them into relation or otherwise), also LPS differs more from my current analysis if odd q are taken into account than CSG', in the sense that CSG' is the same value for q and q/2 when q \(\equiv\) 2 (mod 4) (except in rare events when r=q/2+2), but LPS is not the same value for q and q/2. Conjectures (provocative):  (1) For every \(\varepsilon\)>0, the number of instances where CSG'<0.5\(\varepsilon\) is finite.  (2) The number of instances where CSG'<2/3 is asymptotic to q*whatchamercallit, where whatchamercallit (wc) is a constant in the vicinity of 1/35.  (3) The number of instances where 2/3<=CSG'<=1 is asymptotic to q*wc*gubbins^{(CSG'2/3)}, where gubbins ~ 7*10^{13}.  (4) gubbins^{(1/6)} = wc/3 ?? Unknown:  How does the graph continue past CSG'>1, provided we could calculate a large enough set of samples with larger q? A continuation of (3) for CSG'>1 (assuming it even holds for 2/3<=CSG'<=1) would imply arbitrarily large CSG values for prime gaps in AP, even if it's just for cases where log(p')/log(g) is small. OTOH, assuming a bounded CSG value, the factor by which the number of instances decreases per increment in CSG' has to decrease as well, further down the road.  If (1) is true, then there's a point where asymptotically half of the time either CSG'>0.5+\(\delta\) or CSG'<0.5+\(\delta\). Whether or not \(\delta\) is a constant, who knows? 
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