Re: Do the prime gaps have a limit distribution and what is it?
- --- In email@example.com,
Warren Smith <warren.wds@...> wrote:
> The precise behavior of the extreme record-max gaps alsoIndeed. Measuring average features of a distribution
> (perhaps unfortunately) should be irrelevant to my test.
tells you little about its tail.
Marek Wolf's Fig.1 in
attempts to fit the data on maximal gaps
from Thomas Nicely
and Tomas Oliveira e Silva
and appears to do so quite nicely.
> Marek Wolf's Fig.1 in--Wolf's nonrigorous approximate formula is
> attempts to fit the data on maximal gaps
> from Thomas Nicely
> and Tomas Oliveira e Silva
> and appears to do so quite nicely.
(Max Prime Gap below x) = (x/pi(x)) * (2*ln(pi(x)) - ln(x) + c)
where c = ln(2) + sum(primes p>2) ln( 1 - 1/(p-1)^2 ) = 0.27787688...
It appears to work very well for the first 75 prime gap records.
But Wolf notes his formula conflicts with A.Granville who conjectures
(Max Prime Gap below x) >= 2*exp(-gamma) * ln(x)^2
for infinite set of cases, 2*exp(-gamma)=1.12292.
This Granville conjecture is unsupported by the evidence so far.
- By heuristic arguments I arrived to the simple formula:
Mean Maximum Gap in the neighborhood of N , MMG(N) = [Ln (N) - Ln Ln(N)]^2
This formula represents very well the mean superior boundary of T.Oliveira's cloud of points.
That is, the first occurence of a gap greater than its maximum predecessor is a MMG.
N = 2 4 6 8 14 20 ..... 34 ....
MMG = 3 7 23 89 113 887 ..... 1327....
[Non-text portions of this message have been removed]
- --- In firstname.lastname@example.org, Luis Rodriguez <luiroto@...> wrote:
>--your formula agrees with Marek Wolf's formula at both its dominant and
> By heuristic arguments I arrived to the simple formula:
> Mean Maximum Gap in the neighborhood of N , MMG(N) = [Ln (N) - Ln Ln(N)]^2
> This formula represents very well the mean superior boundary of T.Oliveira's cloud of points.
first subdominant term, as can be shown using claims in