20220730, 08:56  #144  
Dec 2008
you know...around...
853_{10} Posts 
Quote:
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 / gamechanging / boring / etc. 

20220902, 18:53  #145  
Dec 2008
you know...around...
853 Posts 
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.
Quote:
The LHC was worth billions of dollars to find stuff like that in particle physics... 

20220907, 02:11  #146  
Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts 
Quote:
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. 

20220907, 14:41  #147  
Dec 2008
you know...around...
1525_{8} Posts 
Delighted to hear from you again, Robert!
Quote:
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 \(f(g) \cdot (1s(g1)/log(p)) \cdot 2 \cdot c_2/log(p)\). A bit more finetuning should work wonders 

20220909, 13:53  #148 
May 2018
119_{16} Posts 
Is this related to the gaps between nonconsecutive primes? What have you found out about that?

20220909, 16:12  #149 
Dec 2008
you know...around...
853 Posts 
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 maxgap 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. Remotely. This one's more about the nittygritty of the theory of gaps between consecutive primes. 
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