Prime Gap Theory

[ATH]

Sorry, I meant it followed *very* roughly the C-S-G ratio but not in magnitude, I meant in a very non rigorous sence that the current known peak was the same place as for CSG and it was low and very low the same places, i.e. it followed CSG better than the Merit at least. But it would have probably been better saying nothing than just hand waving, good thing RDS is not here.

Here are the updated values even though Robert beat me to it:

Gn	Merit		gn/(ln(pn)^2	Ford-Green-Konyagin-Maynard-Tao		Pn
15900	39.62015365	0.09872683		20.31243105			1.936933265397289504398811903696*10^174
18306	38.06696007	0.07915948		18.72974074			7.041097148478282668812106731813*10^208
10716	36.85828850	0.12677617		20.45427745			1.839377720243795270729953508768*10^126
13692	36.59018324	0.09778276		19.07131098			3.254185929142547441117000456865*10^162
26892	36.42056789	0.04932537		16.38392439			4.696226774889053656642126142794*10^320
66520	35.42445941	0.01886489		13.50196237			3.292808201042179724620296543360*10^815

1476	35.31030807	0.84472754		45.22507308			1425172824437699411
1442	34.97568651	0.84833471		45.29864017			804212830686677669
1454	34.11893253	0.80062005		43.03407437			3219107182492871783
1370	33.76518602	0.83218087		44.30979390			418032645936712127
1132	32.28254764	0.92063859		48.34468117			1693182318746371

4680156	20.37666041	0.00008872		 4.50159088			5.104776509199087740248075761182*10^99749
6582144	13.18288411	0.00002640	 	 2.73531863			8.465069837806447347636518542879*10^216840
5103138	10.22031845	0.00002047	 	 2.12060937			7.695421151871542659687327631743*10^216848

[firejuggler]

So, does that mean that the gap, evaluated with FGKMT valued below 45 (around) are bound for improvement?
What FGKMT value has the CFC 800 gap?

or is it a range thing? ( the higer the gap, the lower the new score is?)

[danaj]

Section 1.1 of the FGKMT paper notes that "These conjectures are well beyond the reach of our methods" with regard to Cramér and Granville.

So far the C-S-G ratio has been a pretty good predictor. One of the interesting things about prime gaps is seeing how well it fits, though that is hampered for large gaps by the sheer magnitude of numbers. The exhaustive search is more useful in this regard.

I, and Dr. Nicely if I infer his meaning in an email he sent me, expect much higher merits to exist at those 120-300 digit sizes, based purely on the CSG values of ~0.1. He notes that in theory there is a 500 digit P1 with a merit of 1000+. Our search methods just aren't good enough (or alternately we need to get busy turning the inner planets into computronium).

So yes, I believe all those first gaps shown -- the highest merits we've found so far -- are doomed to be overtaken with much larger merits. The large ones as well, though those have a different set of computational challenges.

[ATH]

Originally Posted by firejuggler

So, does that mean that the gap, evaluated with FGKMT valued below 45 (around) are bound for improvement?
What FGKMT value has the CFC 800 gap?

or is it a range thing? ( the higer the gap, the lower the new score is?)

You mean the 800 gap from Dr. Nicely's table?:
800 CFC TRNicely 1996 23.66 15 486258341004083
The "FGKMT" constant is 36.75172943

In the paper CRGreathouse linked they say:
G(X) >> log X* loglog X * loglogloglog X / logloglog X
"For any sufficiently large X" and "The implied constant is effective." ??
where G(X) = max(Pn+1 - Pn) for Pn+1 < X

So the maximum prime gap between 1 and X is "much larger" >> than the formula "for any sufficiently large X", I'm not sure if that means that the constant -> infinity for X->infinity ?
But I do not think that means, that because we got a value of 48.34 at 1.7*10^15 there has to be gaps of at least 48 at the 100-200 digit level where the current highest Merit gaps are found.

Regarding the Merit E. Westzynthius proved that the Merit -> infinity as X->infinity, but I assumed that a Merit of 1000+ was out of our current reach at the miliions or billions of digits, but I guess it is possible.

[CRGreathouse]

Originally Posted by ATH

In the paper CRGreathouse linked they say:
G(X) >> log X* loglog X * loglogloglog X / logloglog X
"For any sufficiently large X" and "The implied constant is effective." ??
where G(X) = max(Pn+1 - Pn) for Pn+1 < X

So the maximum prime gap between 1 and X is "much larger" >> than the formula "for any sufficiently large X", I'm not sure if that means that the constant -> infinity for X->infinity ?

>> does not mean "much larger", it is a Vinogradov symbol. Here it means that there are constants G and X0 such that
G(X) > G * log X* loglog X * loglogloglog X / logloglog X
for all X > X0. Informally, >> is "asymptotically at least as large as" (up to a nonzero constant multiple).

"The implied constant is effective" means that not only does such a constant G exist, but that their method makes it possible to compute such a constant (though they have not done so).

They did not prove that the constant can be taken arbitrarily large (I assume this is what you mean by "the constant -> infinity") though 'of course' it can, we just can't prove it.

When you look at large gaps and see worse performance it's because you're comparing the maximal gaps at small sizes to almost-surely-submaximal gaps at large sizes.

Originally Posted by ATH

Regarding the Merit E. Westzynthius proved that the Merit -> infinity as X->infinity

Right, and these results are stronger versions of Westzynthius' theorem.

[mart_r]

I know that the definition of Merit = gap / log(p) = (p₂-p₁) / log(p₁) is deeply immersed in mathematic literature, and the following suggestion only has an impact on small numbers, but:
For my own calculations and considerations, I often use the Gram formula (x) ~ G(x) = 1+

Then, with Merit = G(p₂)-G(p₁), the Cramér-Shanks-Granville ratio would be
(I always found it irritating that CSG can be >1 for the first few gaps, but with the Gram formula, this is not the case.
Does this idea have any supporters?)
Using this method of computation, I made a regression graph for all know CFCs. The x-axis is CSG. Note that the exponent in the regression is close to -2, suggesting that might well be the upper bound for CSG. If instead I let x=1-CSG, the exponent is somewhere around -2.4.
This probably doesn't tell us anything useful, but I just wanted to share some of my thoughts, lest they go to waste.

[robert44444uk]

Originally Posted by mart_r

I know that the definition of Merit = gap / log(p) = (p₂-p₁) / log(p₁) is deeply immersed in mathematic literature, and the following suggestion only has an impact on small numbers, but:
For my own calculations and considerations, I often use the Gram formula (x) ~ G(x) = 1+

Then, with Merit = G(p₂)-G(p₁), the Cramér-Shanks-Granville ratio would be
(I always found it irritating that CSG can be >1 for the first few gaps, but with the Gram formula, this is not the case.
Does this idea have any supporters?)
Using this method of computation, I made a regression graph for all know CFCs. The x-axis is CSG. Note that the exponent in the regression is close to -2, suggesting that might well be the upper bound for CSG. If instead I let x=1-CSG, the exponent is somewhere around -2.4.
This probably doesn't tell us anything useful, but I just wanted to share some of my thoughts, lest they go to waste.

This is interesting, but as usual there will be others who will argue that putting a best fit line in a scatter graph will not necessarily add much insight

I've noticed that large gaps which are not 0mod6 follow a general straight line rule that is evident for example in the post http://www.mersenneforum.org/showpos...39&postcount=7

The graph is reposted here. The spikes are 0mod6 counts of prime gaps in the range. Let us assumed the vertical axis is y and the horizontal x.

The intersect of the straight line at y=0 is of interest. It is dependent on the size of the sample range from which the prime gap count has ben taken - increasing the size of the sample range will increase the value of x at the intersect.

I would conject that all first instance gaps would be located in an area beyond the y=0 intersect and I would suggest that one bounding limit on your graph (the right hand side or top side unless I'm mistaken) would be defined as in terms of x where the straight line intersects with y=0 on my graph. The intersect would be inferred from the xy formula for a sample range close to the p which is the lower prime in the gap.

As can be seen from my graph, gaps beyond the y=0 intersect are super rare, and these freak instances appear to be non-predictable. There were only 14 gaps beyond 640 in the sample range of 230,453,215 gaps and only 7 of these were not 0mod6. None of these large gaps were even remotely close to a first instance gap - for that the testing range would statistically need to be much larger. By increasing the testing range, the y=0 intersect is forced a lot further to the right (i.e. higher x) is but I think the same general observations relating to the randomness of candidates beyond y=0 intersect would still apply.

You should definitely explore this further by taking out gaps that are 0mod 6 from your data set, and refocusing your best fit lines on the bounding limits on the two sides of the scatter data.

[Bobby Jacobs]

I have found a new gap value (g-log²(p)+(2*log(p)*log(log(p))))/log(p). I call it the Jacobs ratio. It appears to have the same distribution on all maximal prime gaps. It is very fun.

[CRGreathouse]

Originally Posted by Bobby Jacobs

I have found a new gap value (g-log²(p)+(2*log(p)*log(log(p))))/log(p). I call it the Jacobs ratio. It appears to have the same distribution on all maximal prime gaps. It is very fun.

Would I be right that you are hinting that the gap between primes is infinitely often as large as

\[
\ell^2 - 2\ell\log\ell
\]

but not larger, up to lower-order terms? (where \(\ell = \log p\))

[Bobby Jacobs]

Yes.

[mart_r]

The title should be turned into a movie, methinks...

Usually we would expect that a gap of at least merit M (=gap/log(p)) appears asymptotically with probability between two consecutive primes p and p+gap. When aiming for gaps with CSG around unity, i.e. M approaching log(p), this approximation becomes distorted and in fact produces wrong results when M>log(p). Case in point: the density of gaps with merit >=M between consecutive primes appears to be on average for integer M with the following values a_M:

M  a_M
1  1.53
2 -1.9
3 -5.3
4 -10
5 -14
6 -19
7 -23
8 -28
9 -32

These are empirical values taken from 2²⁰ prime gaps on either side of e²ⁿ for 30<=2n<=138 (to avoid rounding errors), and the attached graph shows * for 1<=k<=10, with the x-axis indicating 2n. (For individual choices of n, the values are subject to fluctuation depending on the prime factors of n, for example in the vicinity of e³⁰, the value for k=1 is around 2.30, whereas at e³² it's around 0.53. One may also notice that the line for k=3 is a bit more smooth for this kind of reason: only gaps that are a multiple of 6 are compared to each other, leaving hardly distinguishable peaks at multiples of 5 of n.)
* The TEX expression doesn't work and I can't figure out why. Anyway it's in the graph in the attachment.

- Could there be any error in the approach given here? The numbers are relatively stable between e³⁰ and e¹⁰⁰, but tend slightly more to the negative side after that; is my approach still justified? Is there a plausible way to "fix" it if necessary?
- Suppose it could be shown that a_M diverges fast enough, could this be used to argue against Granville's 2e^y log²p? Does it intersect Maier's thesis?
- I've only spent about a day on finding these numbers; is it worth pursuing the calculation further, to obtain some more/more precise numbers? Has someone else already done the number crunching work?
- Is it nonsense what I'm doing here and should I go read some more papers on prime gaps before persecuting any more mindless numerology and asking countless questions?