- 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 - 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

>

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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