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RE Chaotic Series

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  • Jose Ramón Brox
    ... From: Werner D. Sand Let s go out from the alternating series of the odd reciprocals Odd = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +- .
    Message 1 of 3 , Dec 9, 2005
      ----- Original Message -----
      From: "Werner D. Sand" <Theo.3.1415@...>


      Let's go out from the alternating series of the odd reciprocals
      Odd = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +- . -> pi/4 and take only
      the prime fractions: P = -1/3 + 1/5 - 1/7 - 1/11 + 1/13 + 1/17 - 1/19
      - 1/23 + 1/29 - 1/31 + 1/37 . The limit appears to be - 1/3. In this
      case the whole series beginning with 1/5 (P+1/3 = 1/5 - 1/7 .) would
      be = 0, which is rather improbable. Computing 6 million fractions
      results in P ~ - 0.33497. Can someone calculate further?

      ----------------------------------------------------------------------

      gp > k=0;S=0;forprime(n=5,10^M,S=S+(-1)^k*1.0/n;k=k+1);print(S)

      M = 6 (up to 99983, 78496 fractions computed )

      S = 0.1029391841194914003553718838

      M= 8 (up to 99999989, 5761453 fractions computed)

      S = 0.1029396903042634545404718629

      M = 9 (up to 999999937, 50847532 fractions computed )

      S = 0.1029396848049898245067366006

      (If you want to run it yourself, k is the count of fractions computed)

      Regards. Jose Brox
    • Jose Ramón Brox
      ... From: Werner D. Sand Hi José, P is not alternating regularly. ... Sorry, I misread what you wanted. Here are the new calculations:
      Message 2 of 3 , Dec 9, 2005
        ----- Original Message -----
        From: "Werner D. Sand" <Theo.3.1415@...>


        Hi José,

        P is not alternating regularly.

        ---------------------------------------------

        Sorry, I misread what you wanted. Here are the new calculations:

        gp > S=0;forprime(n=5,10^M,S=S+(-1)^((n-1)/2)*1.0/n);print(S)

        M = 4, S = - 0.0003377

        M = 5, S = - 0.00131028

        M = 6, S = - 0.00164580

        M = 7, S = - 0.00163019

        M = 8, S = - 0.00164129937

        M = 9, S = - 0.001645351324

        M = 9.5, S = -0.001647022748

        M = 9.6, S = -0.001647729193

        The last sum is over approximately 1.801*10^8 fractional terms (estimated by the PNT).

        It seems to converge very slowly to - 0.001647... Maybe we could use a convergence
        acceleration method to get more digits and faster; I can't go over M=9.6 with Pari-GP
        (time is not the problem, but primelimit doesn't want to go over 6*10^10).

        Regards. Jose Brox
      • Jose Ramón Brox
        ... From: Jose Ramón Brox It seems to converge very slowly to - 0.001647... ========================================== By the way, it s
        Message 3 of 3 , Dec 9, 2005
          ----- Original Message -----
          From: "Jose Ramón Brox" <ambroxius@...>

          It seems to converge very slowly to - 0.001647...

          ==========================================

          By the way, it's well approximated by -1/607.

          If you add the -1/3, then it's well approximated by -log(57/2)/10.

          Jose
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