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Re: Do the prime gaps have a limit distribution and what is it?

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  • djbroadhurst
    ... Indeed. Measuring average features of a distribution tells you little about its tail. Marek Wolf s Fig.1 in http://arxiv.org/pdf/1010.3945v1.pdf attempts
    Message 1 of 6 , Mar 21, 2012
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      --- In primenumbers@yahoogroups.com,
      Warren Smith <warren.wds@...> wrote:

      > The precise behavior of the extreme record-max gaps also
      > (perhaps unfortunately) should be irrelevant to my test.

      Indeed. Measuring average features of a distribution
      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
    • WarrenS
      ... --Wolf s nonrigorous approximate formula is (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 -
      Message 2 of 6 , Mar 21, 2012
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        > 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.

        --Wolf's nonrigorous approximate formula is

        (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.
      • Luis Rodriguez
        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
        Message 3 of 6 , Mar 22, 2012
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          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]
        • WarrenS
          ... --your formula agrees with Marek Wolf s formula at both its dominant and first subdominant term, as can be shown using claims in
          Message 4 of 6 , Mar 22, 2012
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            --- In primenumbers@yahoogroups.com, Luis Rodriguez <luiroto@...> wrote:
            >
            > 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.

            --your formula agrees with Marek Wolf's formula at both its dominant and
            first subdominant term, as can be shown using claims in
            http://en.wikipedia.org/wiki/Prime-counting_function
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