- 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 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:>

one

> 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

> described by Lagarias and Odlyzko, but uses the Weil explicit formula

18435599767349200867866+3.3823e-08.

> instead of complex curve integrals. The actual value of the analytic

> approximation to pi(10^24) found was

>

10^11

> For the current calculation, all zeros of the zeta function below

> were calculated with an absolute precision of 64 bits.

fftw3-library

>

> 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

> 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 > The actual value of the analytic

Prof. Tomás Oliveira e Silva used 1G zeros, and his estimate of pi(10^24) has a predicted deviation of about 2e6:

> 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.

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

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.

>

> Some tricks were definitely performed there...

Such a method was called Monte-Carlo method by Kevin Stueve:

http://sage.math.washington.edu/edu/2010/414/projects/stueve.pdf