Loading ...
Sorry, an error occurred while loading the content.
 

Forward: pi(10^24) assuming RH

Expand Messages
  • Chris Caldwell
    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
    Message 1 of 5 , Aug 2 10:03 PM
      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
    • 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 2 of 5 , Aug 4 12:50 AM
        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 3 of 5 , Aug 4 12:39 PM
          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
          >




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

          Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          The Prime Pages : http://www.primepages.org/

          Yahoo! Groups Links
        • 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 4 of 5 , Aug 25 8:15 AM
            > 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 5 of 5 , Aug 25 12:12 PM
              > 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
            Your message has been successfully submitted and would be delivered to recipients shortly.