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Re: Riemann Hypothesis

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  • d.broadhurst@open.ac.uk
    ... yes ... sorry John, but you have multiplied by infinity. Neither side is well defined.
    Message 1 of 6 , Jun 28, 2001
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      Jon Perry wrote:
      > pi^2/6 = product(p^2/(p^2-1)
      yes
      > pi^2 . product((p-1)(p+1)) = 6 . product(p^2)
      sorry John, but you have multiplied by infinity.
      Neither side is well defined.
    • S.R.Sudarshan Iyengar
      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
      Message 2 of 6 , May 5, 2002
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        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????


        My advance thanks to you all


        Lots of regards,
        S.R.Sudarshan Iyengar


        [Non-text portions of this message have been removed]
      • Joseph Moore
        AFAIK, the simple relation between primes and the Riemann Hypothesis is the formula: oo ... Zeta(z)= | | ---------- ... p prime Basically, the multiplication
        Message 3 of 6 , May 5, 2002
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          AFAIK, the simple relation between primes and the
          Riemann Hypothesis is the formula:

          oo
          ----- 1
          Zeta(z)= | | ----------
          | | 1-p^(-z)
          p prime

          Basically, the multiplication over all primes... etc.
          I'm sure you are familiar with the formula.

          Now, the above multiplication converging to zero for
          any particular z (Im(z) not zero) is the whole point
          of the hypothesis. In particular, notice that if
          z=-2n+0i for n a natural number, the formula quickly
          converges to zero. These are the trivial zeroes.

          The nontrivial zeroes are expected to lie on the
          complex line z=1/2+bi for some reals b. It is not
          known which subset of the reals or what cardinality
          they fall into.

          The biggest thing about the 'distribution of primes'
          is to dig into the complex algebra a little and find a
          couple things like p^a and cos(b*ln(p)) running around
          (z=a+bi). If the ln function maps most of the primes
          into (for example) 2n*pi < ln(p) < pi+2n*pi, then
          multiplying by a real number b might change that...

          some thoughts

          -jtpk

          --- "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????
          >
          >
          > My advance thanks to you all
          >
          >
          > Lots of regards,
          > S.R.Sudarshan Iyengar
          >
          >
          > [Non-text portions of this message have been
          > removed]
          >
          >


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        • Phil Carmody
          ... hypothesis ... for ... regard???? First hit from the search string Riemann Hypothesis on the Prime Pages: http://primepages.org/notes/rh.html Google also
          Message 4 of 6 , May 5, 2002
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            --- "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

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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 5 of 6 , May 6, 2002
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              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.
              So called "expert" witness for Microsoft. 2002/05/02

              __________________________________________________
              Do You Yahoo!?
              Yahoo! Health - your guide to health and wellness
              http://health.yahoo.com


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