Версія даної теми для друку

Автор: nikelong Feb 21 2009, 20:13

pi(x) project

http://numbers.computation.free.fr/Constants/Primes/Pix/pixproject.html

http://www.rechenkraft.net/wiki/index.php?title=Pi(x)




The pi(x) table project

http://numbers.computation.free.fr/Constants/Primes/Pix/pixtableproject.html



http://www.rechenkraft.net/wiki/index.php?title=Pi(x)_tables

The pi(x) Table Project constructed a very large table of values of pi(x) for large values of x. The table would allow people to study the behavior of the pi(x) function in large ranges, a study which had never before been possible. The project also looked for the first known change of sign of the function pi(x)-Li(x). The first phase of the project computed pi(x) for 1.e16 < x < 1.e17. The last phase computed pi(x) for 1.e17 < x < 1.e18.

Автор: Rilian Feb 22 2009, 20:51

что такое Li(x) ?

Автор: Death Feb 22 2009, 22:24

http://primes.utm.edu/howmany.shtml









Gauss was also studying prime tables and came up with a different estimate (perhaps first considered in 1791), communicated in a letter to Encke in 1849 and first published in 1863.

Gauss:
pi(x) is approximately Li(x) (the principal value of integral of 1/log u from u=0 to u=x).






Below we see an animated image, each successive frame being the result of adding another Tn(x) to the function







http://primes.utm.edu/notes/md6-96.html

Thanks to Paul Zimmermann from INRIA Nancy who lend me some days of computation on his machines, (and also some hours of his time to compile the program on these machines!), and between other values got the pi(418....) = 10^17.

pi(1e19) took 40 hours of computation on a DEC-Alpha 5/250 and needed about 80Mo memory. pi(1e20) took 13days of computation on a DEC-ALPHA 5/250 (because of lack of memory, we had to exchange space against time) and also 13days on a R8000.