Prime Gap Theory

[mart_r]

Originally Posted by Bobby Jacobs

I would give you a like if it was possible to like posts on Mersenne forum.

I count that as a "like". Thanks!

Originally Posted by LaurV

Open the windoze?

It's dark outside as well - but lo! it looks like the dawn breaks on the horizon! Or is it just a faint sheet lightning?

Metaphors aside, I still haven't seen any work related to the bias I'm observing for gaps > m log p. And no clue whether the numbers I'm churning out are wrong / right / biased / not quite correct / interesting / irrelevant / game-changing / boring / etc.

[mart_r]

Hmm. 2,170,000 gaps around 997# show a bias similar to the usual one I found so far, including the effects of the Granville scarcity. The numbers around 2351*2351# I posted earlier are either off because of the extremely small sample size, or larger primorials do have a more serious effect. Maybe it's something else altogether, I dunno, I'm putting these numerical experiments on hiatus for some time.

Originally Posted by mart_r

Craig's data shows that the bias \(f(m) \approx 5 \cdot (m-5/3+O(1))\) for m > 2 (see previous post) - if we assume that the values a_M in post # 139 converge - persists for large merits.
He found 236 gaps >= 1300 between 2⁶⁴ and 2⁶⁴+2^60.023. Usual estimates would predict more than 5000 gaps >= 1300.

m >  count  density    expected   (m+log(density))*log(p)
29.5  176   6.670e-15  1.543e-13  -139.45
30.0   94   3.562e-15  9.358e-14  -145.09
30.5   58   2.198e-15  5.676e-14  -144.33
31.0   36   1.364e-15  3.442e-14  -143.31
31.5   17   6.442e-16  2.088e-14  -154.42
32.0    7   2.653e-16  1.266e-14  -171.61

The values in the last column are analogous to the values a_M mentioned earlier.
a_M or -f(m) or whatever, I'm only talking to myself anyway :)

I've summarized some data to have a more detailed look into small values of m.
There's a peculiar above-average occurrence of gaps with m < 5/3 (ballpark).
(I already suspected Buchstab's function having its share here, though it escapes me how that would be possible.)

Is there at least someone who can reproduce / confirm or even refute these results?

The LHC was worth billions of dollars to find stuff like that in particle physics...

[robert44444uk]

Originally Posted by mart_r

I have found a truly....

For small gaps g and large primes p, the average density of gaps between consecutive primes p_n and p_n+1 is approximately \(f(g) \cdot 2 \cdot c_2/log(p)\), or f(g) times the density of twin (or cousin) primes around p.

Hi Mart

I'm trying to get to grips with what you have written here..but stumbling right at the start

The average density of gaps between consecutive primes....surely there is only one gap between consecutive primes hence trivial.

[mart_r]

Delighted to hear from you again, Robert!

Originally Posted by robert44444uk

Hi Mart

I'm trying to get to grips with what you have written here..but stumbling right at the start

The average density of gaps between consecutive primes....surely there is only one gap between consecutive primes hence trivial.

Yeah, that might've been poorly worded, my mistake. I was hoping it would become clearer during all my explanations :)

For small gaps g and large primes p, the average density of gaps of length g between consecutive primes is approximately \(f(g) \cdot 2 \cdot c_2/log(p)\), or f(g) times the density of twin (or cousin) primes around p.

We get better approximations for g < log(p) by taking \(f(g) \cdot (1-s(g-1)/log(p)) \cdot 2 \cdot c_2/log(p)\). A bit more fine-tuning should work wonders

[Bobby Jacobs]

Is this related to the gaps between non-consecutive primes? What have you found out about that?

[mart_r]

Originally Posted by mart_r

We get better approximations for g < log(p) (...)

I guess I wasn't working at full capacity the other day... I think I meant to cite the approximation formula \(f(g) \cdot e^{-g/log(p)} \cdot 2 \cdot c_2/log(p)\), which works well for all values of g.

But, to summarize again, I've identified three aspects that might go counter to Granville's theory of the largest known prime gaps:
- The density of Granville's sparse intervals where these gaps might occur may themselves be too sparse, specifically they may have logarithmic (or Lebesgue?) measure equal to 0, see my post # 87. Maybe I have messed up the asymptotic boundaries, I'd have to go through it all again to check.
- The aberration connected to the merit of the gap might be an issue, but a linear dependence as mentioned in post # 141 is not enough to work against the quadratic nature of the asymptotic max-gap formula. Maybe there's a second order error term that might help out, but that would be extremely difficult, if not impossible, to find by means of statistical data.
- Said aberration might be more extreme near large primorials. Again, to pin it down would require large sets of data for large numbers, thus a tremendous amount of computation.

Originally Posted by Bobby Jacobs

Is this related to the gaps between non-consecutive primes? What have you found out about that?

Remotely. This one's more about the nitty-gritty of the theory of gaps between consecutive primes.