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RE: [PrimeNumbers] Riemann Hypothesis

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  • Jon Perry
    The big thing with RH is in accurately defining the error term between pi(x), the count of primes to x, and Li(x), the best known approximation, defined by
    Message 1 of 6 , May 6, 2002
      The big thing with RH is in accurately defining the error term between
      pi(x), the count of primes to x, and Li(x), the best known approximation,
      defined by integral(t=0,x, of 1/logt).

      logt is lnt, i.e. the natural logarithm, in base e.

      If RH is true, then we can say that pi(x) ~ Li(x).

      More accurately, if RH is true then we have:

      pi(x) = Li(x) + O(sqrt(x).logx)

      which is a far better error bound that currently known.

      Other connections between RH and the primes include:

      1/zeta(p) is asymtopic to the count of p-free numbers, e.g. if p=5, then
      1/zeta(5) is asymtopic to the count of numbers not involving some q^5.
      (see: http://mathworld.wolfram.com/RiemannZetaFunction.html)

      If psi(x) is the count of primes and prime powers less than x, then the
      zeroes of Riemann's Zeta function can be used to determine psi(x). (almost -
      psi(x) is discontinuous, so RZF defines a modification of psi(x), namely
      psi0(x).
      (see:
      http://users.globalnet.co.uk/~perry/maths/riemannshypothesis/riemannshypothe
      sis.htm)

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com


      -----Original Message-----
      From: Phil Carmody [mailto:thefatphil@...]
      Sent: 06 May 2002 07:44
      To: primenumbers
      Subject: Re: [PrimeNumbers] Riemann Hypothesis



      --- "S.R.Sudarshan Iyengar" <gayathrisr@...> wrote:
      > Dear members,
      >
      > I recently heard that the application of Riemann
      hypothesis
      > will give a hint on the distribution of primes (also, a formula
      for
      > the nth prime can be obtained)???
      >
      > But I am unable to see any relation between Riemann
      > hypothesis and Prime numbers. Can anyone help me in this
      regard????

      First hit from the search string 'Riemann Hypothesis' on the Prime
      Pages:
      http://primepages.org/notes/rh.html

      Google also found:
      http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html

      Phil

      =====
      --
      "One cannot delete the Web browser from KDE without
      losing the ability to manage files on the user's own
      hard disk." - Prof. Stuart E Madnick, MIT.
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