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

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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
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
>
• 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-----
On Behalf Of pbtoau
Sent: Wednesday, August 04, 2010 2:51 AM
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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• ... 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]
• ... 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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