No way to measure record prime gaps

[Bobby Jacobs]

Is there a way to measure how big a record prime gap is compared to the expected record prime gap? It might say how many standard deviations above the expected value the gap is. For example, a value of 0 would be an average record prime gap. A value of 2 would be a surprisingly large record prime gap. A value of -1 would be a small record prime gap. What is a way to measure this?

[ATH]

You take the length of the gap call it: g_n and divide it by ln(p_n) where ln is the natural logarithm. The result is called the merit of the gap:
https://en.wikipedia.org/wiki/Prime_gap

merit = g_n / ln(p_n)

If you get merit > 30 you have a very large gap, if you get merit above 41.938784 you found a record gap.

It is proven that merit can be arbitrarily large as n gets large, so there should be much higher merits our there somewhere.


You can also divide the gap by ln(p_n) twice:
g_n / ln(p_n)²

this is called the Cramér–Shanks–Granville ratio and the record is: 0.9206386 but anything above 0.5 is pretty good if your gap>1500.

The Cramér–Shanks–Granville ratio does not get arbitrarily large, but is conjected to have a maximum somewhere around 1.12

[Bobby Jacobs]

I would really like a measure of record prime gaps where a big value for a record gap between small numbers is a big value for a record gap between large numbers. The size of the gap will not work. For example, the prime gap between 1327 and 1361 has size 34, which is very big for numbers that size. However, a gap of 34 between larger numbers is not very great. The merit of a gap has the same problem. The gap from 1327 to 1361 has merit 4.7, but a merit of 4.7 is not impressive for larger numbers. Even the CSG ratio is not perfect. The CSG ratio of the gap from 1327 to 1361 is 0.65. However, the CSG ratios of the bigger record prime gaps are all at least 0.8. Do you know a prime gap measure where a good value of a record prime gap between small numbers is a good value of a record prime gap between large numbers?

[CRGreathouse]

CSG is approximately scale invariant. A good value at any size will be around 1 (we think), 1327 to 1361 is just a small gap (not a big gap at a low height) in my interpretation. CSG more than 1 + eps for any fixed eps > 0 should be much more rare than merits 'merely' near 1 which should be fairly common (although we haven't seen it happen yet, ignoring 7 and below).

[rudy235]

Originally Posted by Bobby Jacobs

I would really like a measure of record prime gaps where a big value for a record gap between small numbers is a big value for a record gap between large numbers. The size of the gap will not work. For example, the prime gap between 1327 and 1361 has size 34, which is very big for numbers that size. However, a gap of 34 between larger numbers is not very great. The merit of a gap has the same problem. The gap from 1327 to 1361 has merit 4.7, but a merit of 4.7 is not impressive for larger numbers. Even the CSG ratio is not perfect. The CSG ratio of the gap from 1327 to 1361 is 0.65. However, the CSG ratios of the bigger record prime gaps are all at least 0.8. Do you know a prime gap measure where a good value of a record prime gap between small numbers is a good value of a record prime gap between large numbers?

OK once again.
The average gap between 2 primes is approximately ln(p1) -where p1 is the lowest end of the gap-.

This comes from the Prime Number Theorem (PNT) proven by Hadamard and de la Vallée Poussin in the late XIX Century

A gap can in theory be as small as 2 and as big as around ln²p1

Having said that the "merit" which is the ratio of g_n/ln p1 gives an excelent idea of how big is the gap in terms of probability. Merits of 1 are the most common. Merits of 0.5 or of 2 are less frequent. Merits of 0,1 or 10 are even less common and the largest (or smallest) a merit is -relative to 1- the less common it is. It follows more or less a Gaussian distribution with the peak at ln(p).

However (as it has been said before) merits can also be larger than any preset number, if we accept the Cramér's conjecture https://www.dartmouth.edu/~chance/ch...ann/cramer.pdf. For instance in the veciniry of a Googolplex 10^(¹⁰^¹⁰⁰) you should be able to find merits as large as 530

[CRGreathouse]

Well... we should not accept the Cramér conjecture since it's not thought to hold. Mathematicians are divided on what should be true: some think that

is

while others think that it is infinite and a few think it may even be between the two. In the 21st century I've only seen nonmathematicians argue that its value should be 1 and I haven't seen anyone hold that it should be between 1 and 2/e^gamma or less than 1.

[rudy235]

Originally Posted by CRGreathouse

Well... we should not accept the Cramér conjecture since it's not thought to hold. Mathematicians are divided on what should be true: some think that

is

while others think that it is infinite and a few think it may even be between the two. In the 21st century I've only seen nonmathematicians argue that its value should be 1 and I haven't seen anyone hold that it should be between 1 and 2/e^gamma or less than 1.

Hence the "if". (I am not arguing it is correct)

I actually made a big mistake is estimating what the merit of a large gap after a googolplex (a 1 folowed by 10¹⁰⁰ zeroes) could end up being. It would be, again assuming as if the Cramér Conjecture were correct, much larger than 530 and an insane number aproximating 2.3 e 10¹⁰⁰

[ATH]

For the known big prime gaps GSC is only above 0.8 below 2^64 and it gets smaller and smaller with bigger primes even if those gaps have merit above 35, and it is really tiny for the largest gaps.

Any new big gap above 2^64 with GSC above 0.8 or even 0.5 would be something new.

Gn	Pn			Merit		Gn/(ln(Pn)^2)	Ford-Green-Konyagin-Maynard-Tao
5103138	7.69542115*10^216848	10.22031845	0.00002047	 	 2.12060937
6582144	8.46506984*10^216840	13.18288411	0.00002640	 	 2.73531863
4680156	5.10477651*10^99749	20.37666041	0.00008872		 4.50159088
66520	3.29280820*10^815	35.42445941	0.01886489		13.50196237
26892	4.69622677*10^320	36.42056789	0.04932537		16.38392439
26054	5.88832005*10^305	37.00529401	0.05255975		16.80505451
18306	7.04109715*10^208	38.06696007	0.07915948		18.72974074
15900	1.93693327*10^174	39.62015365	0.09872683		20.31243105
13692	3.25418593*10^162	36.59018324	0.09778276		19.07131098
10716	1.83937772*10^126	36.85828850	0.12677617		20.45427745
8382	1.74442287*10^96	37.82412584	0.17068295		22.59742318
8350	2.93703234*10^86	41.93878373	0.21064211		25.84973884
1510	6787988999657777797	34.82336886	0.80309074		43.33266457
1454	3219107182492871783	34.11893253	0.80062005		43.03407437
1476	1425172824437699411	35.31030807	0.84472754		45.22507308
1442	804212830686677669	34.97568651	0.84833471		45.29864017
1370	418032645936712127	33.76518602	0.83218087		44.30979390
1132	1693182318746371	32.28254764	0.92063859		48.34468117

Ford-Green-Konyagin-Maynard-Tao:

        log X * loglog X * loglogloglog X
G(X) >  ---------------------------------
                  logloglog X

[henryzz]

Do we have any idea of the probability distribution for CSG or any of the other gap rating systems?
Are there any that are independent of the size of p_n?

[CRGreathouse]

Originally Posted by henryzz

Do we have any idea of the probability distribution for CSG or any of the other gap rating systems?
Are there any that are independent of the size of p_n?

g_n/log(p_n) is independent of the size of p_n -- it's approximately exponentially exponentially distributed with λ = log 2. But I don't think that's what you mean.

[ATH]

Looking at Dr, Nicely's list there are many gaps between 2^64 and 10^27 with GSC above 0.5, the highest GSC I found so far above 2^64 is:
1750 C?C Spielaur 2016 32.10 24 475135024904107611376237

with GSC=0.58878947