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Re: Forward: pi(10^24) assuming RH

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  • pbtoau
    Congratulations on this accomplishment! I am looking forward to the paper describing the work. Is there a pre-print available? I do have two questions about
    Message 1 of 5 , Aug 4, 2010
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      Congratulations on this accomplishment! I am looking forward to the paper describing the work. Is there a pre-print available? I do have two questions about what has been reported so far. The value of pi(10^24) is a 74-bit number. How can it be calculated using 64-bit arithmetic? I was under the impression that calculating pi(10^n) required the zeros less than 10^n/2. There are 357948363084 zeros less than 10^11. This is 1/11th of the 3945951430271 zeros less than 10^12. Proving that you only need that many fewer zeros to settle on the correct value for pi(10^24) is quite an accomplishment. Since the fractional portion of the approximation is so small it is even more remarkable. I wonder how many zeros would have been sufficient to establish this new record. Best regards, David Baugh


      --- In primenumbers@yahoogroups.com, "Chris Caldwell" <caldwell@...> wrote:
      >
      > From: Jens Franke
      > Sent: Thursday, July 29, 2010 2:47 PM
      > Subject: pi(10^24)
      >
      > Using an analytic method assuming (for the current calculation) the
      > Riemann Hypthesis, we found that the number of primes below 10^24 is
      > 18435599767349200867866. The analytic method used is similar to the one
      > described by Lagarias and Odlyzko, but uses the Weil explicit formula
      > instead of complex curve integrals. The actual value of the analytic
      > approximation to pi(10^24) found was 18435599767349200867866+3.3823e-08.
      >
      > For the current calculation, all zeros of the zeta function below 10^11
      > were calculated with an absolute precision of 64 bits.
      >
      > We also verified the known values of pi(10^k) for k<24, also using the
      > analytic method and assuming the Riemann hypothesis.
      >
      > Other calculations of pi(x) using the same method are (with the
      > deviation
      > of the analytic approximation from the closest integer included in
      > parenthesis)
      >
      > pi(2^76)=1462626667154509638735 (-6.60903e-09)
      > pi(2^77)=2886507381056867953916 (-1.72698e-08)
      >
      > Computations were carried out using resources at the Institute for
      > Numerial Simulation and the Hausdorff Center at Bonn University. Among
      > others, the programs used the GNU scientific library, the fftw3-library
      > and mpfr and mpc, although many time critical floating point
      > calculations
      > were done using special purpose routines.
      >
      > J. Buethe
      > J. Franke
      > A. Jost
      > T. Kleinjung
      >
    • Chris Caldwell
      Jens Franke (Universitat Bonn, http://www.math.uni-bonn.de/members?mode=struc) is not on this list. I just thought the item might interest some who are so
      Message 2 of 5 , Aug 4, 2010
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        Jens Franke (Universitat Bonn,
        http://www.math.uni-bonn.de/members?mode=struc) is not on this list.

        I just thought the item might interest some who are so forwarded it. CC

        -----Original Message-----
        From: primenumbers@yahoogroups.com [mailto:primenumbers@yahoogroups.com]
        On Behalf Of pbtoau
        Sent: Wednesday, August 04, 2010 2:51 AM
        To: primenumbers@yahoogroups.com
        Subject: [PrimeNumbers] Re: Forward: pi(10^24) assuming RH

        Congratulations on this accomplishment! I am looking forward to the
        paper describing the work. Is there a pre-print available? I do have
        two questions about what has been reported so far. The value of
        pi(10^24) is a 74-bit number. How can it be calculated using 64-bit
        arithmetic? I was under the impression that calculating pi(10^n)
        required the zeros less than 10^n/2. There are 357948363084 zeros less
        than 10^11. This is 1/11th of the 3945951430271 zeros less than 10^12.
        Proving that you only need that many fewer zeros to settle on the
        correct value for pi(10^24) is quite an accomplishment. Since the
        fractional portion of the approximation is so small it is even more
        remarkable. I wonder how many zeros would have been sufficient to
        establish this new record. Best regards, David Baugh


        --- In primenumbers@yahoogroups.com, "Chris Caldwell" <caldwell@...>
        wrote:
        >
        > From: Jens Franke
        > Sent: Thursday, July 29, 2010 2:47 PM
        > Subject: pi(10^24)
        >
        > Using an analytic method assuming (for the current calculation) the
        > Riemann Hypthesis, we found that the number of primes below 10^24 is
        > 18435599767349200867866. The analytic method used is similar to the
        one
        > described by Lagarias and Odlyzko, but uses the Weil explicit formula
        > instead of complex curve integrals. The actual value of the analytic
        > approximation to pi(10^24) found was
        18435599767349200867866+3.3823e-08.
        >
        > For the current calculation, all zeros of the zeta function below
        10^11
        > were calculated with an absolute precision of 64 bits.
        >
        > We also verified the known values of pi(10^k) for k<24, also using the

        > analytic method and assuming the Riemann hypothesis.
        >
        > Other calculations of pi(x) using the same method are (with the
        > deviation
        > of the analytic approximation from the closest integer included in
        > parenthesis)
        >
        > pi(2^76)=1462626667154509638735 (-6.60903e-09)
        > pi(2^77)=2886507381056867953916 (-1.72698e-08)
        >
        > Computations were carried out using resources at the Institute for
        > Numerial Simulation and the Hausdorff Center at Bonn University. Among

        > others, the programs used the GNU scientific library, the
        fftw3-library
        > and mpfr and mpc, although many time critical floating point
        > calculations
        > were done using special purpose routines.
        >
        > J. Buethe
        > J. Franke
        > A. Jost
        > T. Kleinjung
        >




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      • Andrey Kulsha
        ... Prof. Tomás Oliveira e Silva used 1G zeros, and his estimate of pi(10^24) has a predicted deviation of about 2e6: http://www.ieeta.pt/~tos/primes.html#e
        Message 3 of 5 , Aug 25, 2010
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          > The actual value of the analytic
          > approximation to pi(10^24) found was 18435599767349200867866+3.3823e-08.
          >
          > For the current calculation, all zeros of the zeta function below 10^11
          > were calculated with an absolute precision of 64 bits.

          Prof. Tomás Oliveira e Silva used 1G zeros, and his estimate of pi(10^24) has a predicted deviation of about 2e6:
          http://www.ieeta.pt/~tos/primes.html#e

          So with 358G zeros the Weil's formula would give the deviation of about 1e5, but not as small as 3e-8.

          Some tricks were definitely performed there...

          Waiting for the paper :-)

          Andrey

          [Non-text portions of this message have been removed]
        • andrey_601
          ... One of them could be evaluation of pi(x) for the large number of samples near x = 10^24 (maybe that s why fftw3 was used). After some sieving we obtain
          Message 4 of 5 , Aug 25, 2010
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            > So with 358G zeros the Weil's formula would give the deviation of about 1e5, but not as small as 3e-8.
            >
            > Some tricks were definitely performed there...

            One of them could be evaluation of pi(x) for the large number of samples near x = 10^24 (maybe that's why fftw3 was used). After some sieving we obtain many estimations for pi(10^24) and take the mean of them, reducing the error statistically.

            Such a method was called Monte-Carlo method by Kevin Stueve:
            http://sage.math.washington.edu/edu/2010/414/projects/stueve.pdf
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