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Fwd: [PrimeNumbers] Estimating the density of Primes

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  • Punkish301@aol.com
    In a message dated 20/05/01 02:23:59 GMT Daylight Time, Punkish301 writes:
    Message 1 of 2 , May 19, 2001
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      In a message dated 20/05/01 02:23:59 GMT Daylight Time, Punkish301 writes:

      << In a message dated 20/05/01 01:59:28 GMT Daylight Time,
      tech_newsletters@... writes:
      Dear Michael,
      <<
      Do we know what the value of a is?
      Since Riemann's hypothesis is generally believe to be true, should
      one use that value of pi(x)?
      Li(x) <=> integral from 0 to x of x/log x, right?
      What does "O(z)" mean as it is used in the statements of pi(x) above?
      How would one use the PNT or its improved version to determine the
      density of Primes of particular forms over arbitrary ranges? (fixed
      k, fixed n, or variable k,n)
      >>
      Considering you've been looking for primes for less than a month, you've
      done well understanding a bit of PNT, which I havent really got to grips with
      after five years...lol.

      Couple of things, the integral Li(x) is 1/log t dt between 0 and x, rather
      than x/log x, and I'm not really sure if mathematicians would be so
      interested in the value of the constant A in the error term of PNT - and what
      exactly would we learn if we did? We know the error term and that is what
      counts (do people on the board agree here, or not? I might not have defined
      the integral too well...sorry)

      O(z) - I think O(x*exp(-A*sqrt(log(x))) means "of the order of magnitude of
      this formula" ie the average of Li(x) minus pi(x), it's meant to be taken as
      x tends to infinity, not for a particular value of x - I think (will anyone
      like to correct me if I'm wrong).

      I don't think mathematicians like to rely too heavily on Riemann, since it's
      not known conclusively - nothing concrete yet, despite lots of evidence by
      various people that it *might* be true, there still might exist a complex
      zero that doesn't fit the conjecture which would spoil the whole thing! - and
      I don't understand what you mean by "should one use that value of pi(x)" can
      you explain this a bit?

      I'll leave the rest to Yves...he knows more about the other formulae than I
      do,
      Thanks, from Guy >>



      [Non-text portions of this message have been removed]
    • tech_newsletters@yahoo.com
      ... I can t saw I truely understand it, but I am trying since I am a prospective math major :) ... Ah, So the O( ... ) is for Order of magnitude like in
      Message 2 of 2 , May 19, 2001
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        --- In primenumbers@y..., Punkish301@a... wrote:
        > In a message dated 20/05/01 02:23:59 GMT Daylight Time, Punkish301
        writes:
        >
        > << In a message dated 20/05/01 01:59:28 GMT Daylight Time,
        > tech_newsletters@y... writes:
        > Dear Michael,
        > <<
        > Do we know what the value of a is?
        > Since Riemann's hypothesis is generally believe to be true,
        > should one use that value of pi(x)?
        > Li(x) <=> integral from 0 to x of x/log x, right?
        > What does "O(z)" mean as it is used in the statements of pi(x)
        > above? How would one use the PNT or its improved version to
        > determine the density of Primes of particular forms over
        > arbitrary ranges? (fixed k, fixed n, or variable k,n)
        > >>
        > Considering you've been looking for primes for less than a month,
        > you've done well understanding a bit of PNT, which I havent really
        > got to grips with after five years...lol.

        I can't saw I truely understand it, but I am trying since I am a
        prospective math major :)

        > O(z) - I think O(x*exp(-A*sqrt(log(x))) means "of the order of
        > magnitude of this formula" ie the average of Li(x) minus pi(x),
        > it's meant to be taken as x tends to infinity, not for a
        > particular value of x - I think (will anyone like to correct me if
        > I'm wrong).

        Ah, So the O( ... ) is for Order of magnitude like in complexity
        analysis, which programmers among other use.

        > I don't think mathematicians like to rely too heavily on Riemann,
        > since it's not known conclusively - nothing concrete yet, despite
        > lots of evidence by various people that it *might* be true, there
        > still might exist a complex zero that doesn't fit the conjecture
        > which would spoil the whole thing!

        Of course, but I had gotten the impression that the Riemann's
        hypothesis is generally believed to be true, by the same token as
        Fermat's Last Theorem was believed to be true in the early 1990's.
        Never the less, I concur that we should not 'count' on it to be true.
        Would it be wise to use it in a probability calculation?

        > I don't understand what you mean by "should one use that value of
        > pi (x)" can you explain this a bit?

        Yes, I was unclear. I meant should one use pi(x) = Li(x) + O(x^(1/2)
        log x) to derive a density function for primes. Since the O(x) is
        just the error term, my question is probably mute, unless one would
        need to use the error term when making a particularly accurate
        statement of the probability of finding prime(s) in a range.



        Additionally I am assuming that the formula Gallot references for
        determining the density of prime(s) is derived in some way from the
        Prime Number Theorem (PNT). Is this a valid deduction?
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