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Re: sum(1/(p(i)*log(p(i))), i=1...infinity)=Pi/2 ????

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  • David Broadhurst
    ... Certainly not. I believe that the sum exceeds 1.6366. To see this, do the sum over primes p
    Message 1 of 4 , Jan 12, 2009
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      --- In primenumbers@yahoogroups.com, "AlexP" <apovolot@...> wrote:

      > sum(1/(p(i)*log(p(i))), i=1...infinity)=Pi/2 ????

      Certainly not. I believe that the sum exceeds 1.6366.

      To see this, do the sum over primes p <= N explicitly and then use
      the prime number theorem to estimate the residuum as 1/log(N).

      S(N)=local(s=0);forprime(p=2,N,s=s+1/(p*log(p)));s+1/log(N);
      for(k=6,9,print(S(10^k)))

      1.63661864
      1.63661681
      1.63661652
      1.63661639

      Perhaps Andrey Kulsha can provide a better estimate?

      David Broadhurst
      ----------------------------------------------------------------
      The Open University is incorporated by Royal Charter (RC 000391),
      an exempt charity in England and Wales and
      a charity registered in Scotland (SC 038302).
    • David Broadhurst
      ... Steve Finch gives the value ... in http://tinyurl.com/9x29z2 David Broadhurst ... The Open University is incorporated by Royal Charter (RC 000391), an
      Message 2 of 4 , Jan 12, 2009
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        --- In primenumbers@yahoogroups.com, "David Broadhurst"
        <d.broadhurst@...> wrote:

        > do the sum over primes p <= N explicitly and then use
        > the prime number theorem to estimate the residuum as 1/log(N)
        ...
        > 1.63661639

        Steve Finch gives the value

        > 1.6366163233...

        in http://tinyurl.com/9x29z2

        David Broadhurst
        ----------------------------------------------------------------
        The Open University is incorporated by Royal Charter (RC 000391),
        an exempt charity in England and Wales and
        a charity registered in Scotland (SC 038302).
      • Andrey Kulsha
        ... http://pi.lacim.uqam.ca/eng/table_en.html http://pi.lacim.uqam.ca/piDATA/plogp.txt ... Best, Andrey
        Message 3 of 4 , Jan 12, 2009
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          > Perhaps Andrey Kulsha can provide a better estimate?

          http://pi.lacim.uqam.ca/eng/table_en.html
          http://pi.lacim.uqam.ca/piDATA/plogp.txt

          :-)

          Best,

          Andrey
        • Andrey Kulsha
          ... And http://www.research.att.com/~njas/sequences/A137245 of course. Best, Andrey
          Message 4 of 4 , Jan 12, 2009
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