20180520, 23:19  #23 
May 2018
233 Posts 
Here are some values of some prime gap measures. The first prime is p_{1}, the second prime is p_{2}, and the gap is g=p_{2}p_{1}.
p_{1}: 2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009 g: 1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282 p_{2}: 3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291 g/ln(p_{1}): 1.442695041, 1.820478453, 2.055593370, 1.913573933, 1.782278478, 2.961466361, 2.875591620, 2.946443246, 3.129851463, 4.728345408, 3.928243586, 4.554708602, 5.261164230, 6.953520217, 7.192376566, 7.502537053, 8.735011648, 8.697998417, 8.359741402, 9.347822898, 10.19704418, 10.03068891, 10.80966760, 12.46145233, 12.44865987, 11.92209966, 12.27641993, 13.00297965, 12.64274696, 14.17528581 g/ln(p_{2}): 0.9102392264, 1.242669869, 1.668129566, 1.781845226, 1.748744355, 2.890061790, 2.860130432, 2.936796132, 3.121281768, 4.711767984, 3.926631632, 4.553388044, 5.259754890, 6.951982323, 7.192044900, 7.502381009, 8.734805614, 8.697844701, 8.359689617, 9.347758521, 10.19699247, 10.03066729, 10.80966075, 12.46144487, 12.44865659, 11.92209850, 12.27641914, 13.00297877, 12.64274655, 14.17528535 g/ln^{2}(p_{1}): 2.081368981, 1.657070898, 1.056366025, 0.6102942000, 0.3970645718, 0.6264487862, 0.4593903979, 0.4340763900, 0.4452713720, 0.6575661849, 0.4286416021, 0.4714856920, 0.5323047896, 0.6715478251, 0.6015148915, 0.5863339816, 0.6812538258, 0.6636418989, 0.5922481044, 0.6619832798, 0.7025656074, 0.6533423380, 0.6491606315, 0.7394656865, 0.7044051476, 0.6402543257, 0.6440619075, 0.6817640317, 0.6393562030, 0.7125486806 g/ln^{2}(p_{2}): 0.8285354492, 0.7721142022, 0.6956640620, 0.5291620684, 0.3822633524, 0.5966040821, 0.4544636715, 0.4312385758, 0.4428363584, 0.6529634570, 0.4282898882, 0.4712123337, 0.5320196440, 0.6712508086, 0.6014594167, 0.5863095918, 0.6812216886, 0.6636184425, 0.5922407668, 0.6619741618, 0.7025584821, 0.6533395208, 0.6491598082, 0.7394648018, 0.7044047769, 0.6402542013, 0.6440618244, 0.6817639395, 0.6393561615, 0.7125486341 (gln^{2}(p_{1}))/ln(p_{1}): 0.7495478602, 0.7218661633, 0.1096832205, 1.221920283, 2.706357891, 1.765921458, 3.383989844, 3.841401736, 3.899236100, 2.462330627, 5.236157553, 5.105624000, 4.622579692, 3.400947413, 4.764728184, 5.293134509, 4.086951795, 4.408465219, 5.755527761, 4.773112153, 4.316965716, 5.322194759, 5.842093269, 4.390516001, 5.223925161, 6.698781420, 6.784511621, 6.069571917, 7.131436667, 5.718492958 (gln^{2}(p_{2}))/ln(p_{2}): 0.1883730631, 0.3667680428, 0.7297657073, 1.585450605, 2.825966624, 1.954125295, 3.433288847, 3.873346319, 3.927104640, 2.504207017, 5.241531661, 5.109746216, 4.626637369, 3.404775885, 4.765611257, 5.293556689, 4.087460266, 4.408850569, 5.755666980, 4.773273778, 4.317091027, 5.322249480, 5.842110685, 4.390533533, 5.223933089, 6.698784388, 6.784513638, 6.069574086, 7.131437716, 5.718494068 The measure (gln^{2}(p_{1}))/ln(p_{1}) does not have the same distribution for all maximal gaps because it is more negative for bigger numbers. However, if we have a better formula for G(p) than ln^{2}(p), then it will have the same distribution. What is a better formula than ln^{2}(p)? 
20180603, 23:28  #24 
May 2018
E9_{16} Posts 
Hi! Does anyone know a formula for the expected maximal prime gap for primes up to p?
Last fiddled with by Bobby Jacobs on 20180603 at 23:29 
20180604, 00:29  #25 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 

20180604, 14:34  #26  
Einyen
Dec 2003
Denmark
5^{2}×127 Posts 
Quote:
https://arxiv.org/abs/1408.5110 Quote:
https://www.youtube.com/watch?v=BH1GMGDYndo&t=7m44s 

20180604, 14:43  #27 
Aug 2006
3×1,993 Posts 
Bobby, I'm not sure that I understand your objection to the standard Perhaps if you could explain in what way you feel this does not meet your needs it would help us find something more suitable for your situation.

20180605, 22:58  #28 
May 2018
11101001_{2} Posts 
If G(p) is the expected maximal prime gap between primes up to p, then (gG(p))/ln(p) should be a measure with the same distribution on all maximal prime gaps. The measure (gln^{2}(p))/ln(p) was not like that. There should be a better approximation than ln^{2}(p).

20180606, 12:13  #29 
Aug 2006
1011101011011_{2} Posts 

20180606, 13:25  #30 
Feb 2017
Nowhere
3×1,657 Posts 

20180607, 22:23  #31 
May 2018
E9_{16} Posts 

20180617, 22:15  #32 
May 2018
11101001_{2} Posts 
There is also something called the gap value. It is sqrt(g)/ln(p_{1}+p_{2}). Here are the gap values of the first 30 maximal prime gaps.
0.6213349347, 0.6800929643, 0.6919525124, 0.6199287913, 0.5412484188, 0.6827045970, 0.6087185235, 0.5969053332, 0.6066293050, 0.7384173974, 0.6085543782, 0.6405915434, 0.6816936284, 0.7679848950, 0.7330617929, 0.7263691872, 0.7830410800, 0.7737166488, 0.7335523374, 0.7755517622, 0.7999848732, 0.7733788478, 0.7735067688, 0.8259491127, 0.8076125106, 0.7714425767, 0.7743746631, 0.7967345566, 0.7725183220, 0.8157048503 This is similar to the CSG ratio because it approaches 1 with bigger maximal gaps. Last fiddled with by Bobby Jacobs on 20180617 at 22:18 
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