- --- In primenumbers@yahoogroups.com,

Warren Smith <warren.wds@...> wrote:

> The precise behavior of the extreme record-max gaps also

Indeed. 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

http://arxiv.org/pdf/1010.3945v1.pdf

attempts to fit the data on maximal gaps

from Thomas Nicely

http://www.trnicely.net

and Tomas Oliveira e Silva

http://www.ieeta.pt/~tos/gaps.html

and appears to do so quite nicely.

David > Marek Wolf's Fig.1 in

--Wolf's nonrigorous approximate formula is

> http://arxiv.org/pdf/1010.3945v1.pdf

> attempts to fit the data on maximal gaps

> from Thomas Nicely

> http://www.trnicely.net

> and Tomas Oliveira e Silva

> http://www.ieeta.pt/~tos/gaps.html

> 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.

Example:

N = 2 4 6 8 14 20 ..... 34 ....

MMG = 3 7 23 89 113 887 ..... 1327....

Ludovicus

[Non-text portions of this message have been removed] - --- In primenumbers@yahoogroups.com, 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

http://en.wikipedia.org/wiki/Prime-counting_function